,SON,  HINKLE   &   CO, 
NEW  YORK:  CLARK  &  MAYNARD. 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Received      ^Le  ,  189  9- 

Accession  No.  6  X/~«/~"4?    .    Cla&s  No. 


WHITE'S   GIIADED-SCIIOOL   SERIES. 


AN 


INTERMEDIATE 


ARITHMETIC, 


MENTAL  AND  WRITTEN  EXERCISES 


NATUKAL  SYSTEM  OF  INSTRUCTION. 


ILSOIN-,     H  I  ]S~  K  L  E     &     CO. 

PHII/A:    CLAXTOM,  REMSEN  &  HAFFELFINGER. 
NEW  YORK:    CLARK  &  MAYNARD. 


PUBLISHERS'  NOTICE. 


WHITE'S   GRADED-SCHOOL   SERIES. 

Complete  in  Three  Boohs  : 

I.   PRIMARY  ARITHMETIC. 
II.   INTERMEDIATE  ARITHMETIC. 
III.    COMPLETE  ARITHMETIC. 

o<>^jt;oo — 

This  Series  of  Arithmetics  is  specially  designed  for  Graded 
Schools,  the  successive  books  being  respectively  adapted,  both  in 
matter  and  method,  to  the  several  grades  of  pupils  using  them. 
Neither  book  is  an  epitome  of  the  succeeding  one. 

The  Series  is  the  only  one,  yet  published,  which  combines  Mental 
and  Written  Arithmetic  in  a  practical  and  philosophical  manner. 
The  two  classes  of  exercises  go  hand  in  hand  throughout  the  Series, 
each  being  made  the  complement  of  the  other. 

The  Series  also  faithfully  embodies  the  Inductive  Method  of  In- 
struction. The  definitions,  principles,  and  rules  are  placed  after 
the  problems,  and  are  deduced  from  the  processes. 

These  three  important  features  have  permitted  the  presentation 
of  the  whole  subject  of  Arithmetic  in  much  less  space  than  is  em- 
ployed iii  other  series.  The  use  of  White's  Graded-School  Arith- 
metics will  result  in  a  mastery  of  this  branch  in  full  ONE-THIRD 
less  time  than  is  now  devoted  to  it. 


7 


Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

WILSON,  HINKLE  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for 
the  Southern  District  of  Ohio. 

ELECTROTYPED  AT  THE  FRANKLIN  TYPE  FOUNDRY,   CINCINNATI. 


PEEFACE. 


IT  is  claimed  for  this  treatise  that  it  possesses  three  very 
important  characteristics,  to  wit: 

1.  It  is  specially  adapted  to  the  grade  of  pupils  for  which  it 
is  designed.     It  is  not  an  abridgment  of  the  Complete  Arith- 
metic.    It  presents    only  those   operations   and   principles 
which  can  be  mastered  by  intermediate  classes,  and  each 
subject  is  treated  as  thoroughly  as  the  advancement  of  the 
pupils  will  permit.     It  is  also  believed  that  the  subjects  are 
introduced  in  the  best  possible  order.     There  are  reasons  in 
favor  of  placing  United  States  Money  before  Fractions,  but 
stronger  reasons  favor  the  reverse  order  of  arrangement  in 
this  work. 

2.  It  combines  Mental  and  Written  Arithmetic  in  a  practical 
and  philosophical  manner.     This  is  done  by  making  every 
mental  exercise  preparatory  to    a  written   one;    and  thus 
these  two  classes  of  exercises,  which  have  been  so  unnatu- 
rally divorced,  are  united  as  the  essential  complements  of 
each  other.    This  union  is  natural  and  complete,  and,  as  a 
consequence,  the  several  subjects  are  treated  in  much  less 
space  than   is  possible  when  mental  and  written  exercises 
are  presented  in  separate  books. 

3.  It  faithfully  embodies  -the  Inductive  Method.     Instead  of 
attempting  to  deduce  a  principle  or  rule  from  a  single  ex- 
ample, as  is  usually  done,  each  process  is  developed  induct- 
ively,  and   the   successive   steps   are   thoroughly  mastered 
and  clearly  stated  by  the  pupil  before  he  is  confronted  with 
the  author's  generalization.     See  "Suggestions  to  Teachers." 
This  method  not  only  places  "processes  before  rules,"  but 


IV  PRKFAGE. 

it  teaches  "  rules  through  processes,"  thus  observing  two 
important  inductive  maxims.  The  work  also  observes  the 
inductive  method  in  not  giving  answers  to  questions  and 
problems.  For  an  illustration,  see  the  method  used  in  de- 
veloping the  idea  of  a  Fraction  (p.  94).  Printed  answers  to 
the  questions  there  asked,  would  clearly  violate  the  wise 
maxim:  "Never  tell  a  pupil  any  thing  which  you  can  lead 
him  to  discover  and  express  for  himself." 

Attention  is  also  called  to  the  use  of  visible  illustrations 
(objects  or  pictures)  in  developing  new  ideas  and  processes. 
In  the  fundamental  rules  this  illustrative  or  perceptive  step 
is  omitted,  since  it  is  fully  presented  in  the  Primary  Arith- 
metic. The  engraved  cuts  in  Fractions,  United  States 
Money,  and  Denominate  Numbers,  are  specially  designed  to 
be  used  as  a  means  of  developing  and  illustrating  the  sub- 
jects considered.  It  is  hoped  that  they  may  be  found  as 
useful  as  they  are  beautiful. 

Two  other  features,  worthy  of  special  notice,  are  the  great 
variety  of  exercises,  and  their  preeminently  progressive  character. 
Generally,  each  lesson  contains  both  concrete  and  abstract 
examples,  and  every  new  process  or  combination  is  at  once 
used  in  the  solution  of  problems  involving  mental  analysis. 
This  arrangement  avoids  the  mechanical  monotony  which 
characterizes  long  drills  on  a  single  class  of  exercises.  The 
problems,  all  of  which  are  original,  are  so  graded  that  they 
present  but  one  difficulty  at  a  time,  and  all  in  their  natural 
order.  The  pupil's  progress  is  thus  made  easy  and  thorough. 

It  is  hoped  that  these  and  other  features  may  commend 
this  work  to  all  progressive  teachers,  and  that  it  may  prove 
as  successful  in  the  school-room  as  its  plan  is  natural  and 
simple. 

Columbus,  Ohio,  May,  1870. 


SUGGESTIONS  TO  TEACHERS. 


IN  the  preparation  of  this  work  two  facts  were  kept  in 
view,  viz.:  (1)  that  it  is  to  be  studied  by  pupils  who  must 
largely  depend  upon  the  living  teacher  for  explanations,  and 
(2)  that  those  methods  which  are  most  natural  and  simple, 
are  most  successful  in  practice.  Hence,  its  pages  are  not 
cumbered  with  long  verbal  explanations  and  peculiar  meth- 
ods, of  little  practical  use  to  pupil  or  teacher.  The  author 
has  left  something  for  the  teacher  to  do,  and  that  this  may  be 
done  wisely,  he  offers  the  following  hints  and  suggestions : 

1.  Mental  Exercises. — These   exercises   should   be  made  a 
thorough  intellectual  drill.    They  should  be  recited  mentally, 
that  is,  without  writing  the  results,  and,  since  the  reasoning 
faculty  is  not  trained  by  logical  verbiage,  the  solutions  should 
be  concise  and  simple.     See  pages  25,  97,  etc.     They  should 
also  be  made  introductory  to  the  Written  Exercises,  of  which 
they  are  often  a  complete  elucidation.     The  corresponding 
examples  in  the  two  classes  of  exercises  should  be  recited 
together  as  well  as  separately. 

2.  Written  Exercises. — The  pupils  should  be  required  to 
solve  every  problem  of  the  assigned  lesson  on  the  slate  or 
paper,  and  the  solutions  should  be  brought  to  the  recitation 
for  the  teacher's  inspection  and  criticism.     Since  the  answers 
are  not  given  in  the  book,  they  should  be  obtained  by  a  com- 
parison of  the  pupils'  results.     From  three  to  five  minutes 
will  suffice  to  ascertain  the  answers  to  twenty  problems,  and 
also  to  test  the  accuracy  and  neatness  of  each  pupil's  work. 
The  time  thus  employed  is  more  than  made  good  by  the  in- 
creased interest,  self-reliance,  and  study  which  the  absence 
of  answers  secures.    The  mental  problems  may  also  be  solved 
on  the  slate  or  paper  in  preparing  the  lesson,  and  then  re- 
cited, not  only  mentally  as  above  described,  but  also  as  a 
written  exercise.     This  will  increase  the  number  of  written 
problems,  and,  at  the  same  time,  it  will  secure  a  careful 
preparation  of  the  entire  lesson. 

3.  Definitions  and    Principles. — These  should   be  deduced 
and  stated  by  the  pupils  under  the  guidance  of  the  teacher, 
and  usually  in   connection  with  the  solution  of  problems. 
Take  for  illustration  the  definition  of  multiplication.     The 
pupils  multiply  304  by  5.     The  teacher  asks,  What  have  you 
done?     "I  have  multiplied  304  by  5."     T.  Do  not  use 'the 
word  "  multiplied."     (If  necessary  the  teacher  shows  what  is 
meant  by  taking  a  thing  one  or  more  times.)     "  I  have  taken 
304  five  times."     T.  By  what  process  have  you  taken  304  fivo 

(v) 


VI  SUGGESTIONS    TO    TEACHERS. 

times?  "By  multiplying  it."  T.  What  then  is  multiplica- 
tion? "Multiplication  is  the  process  of  taking  a  number 

."     T.  How  many  times  is  the   number  taken   in  the 

above  example?  "It  is  taken  five  times,  or  as  many  times 
as  there  are  units  in  the  multiplier."  T.  Now  complete 
your  definition.  "Multiplication  is  the  process  of  taking  a 
number  as  many  times  as  there  are  units  in  another  number" 
These  steps  should  be  repeated  with  other  examples  until 
the  definition  is  clearly  reached  and  accurately  stated.  It 
should  then  be  written  and  compared  with  the  author's  defi- 
nition, which  should  be  thoroughly  memorized. 

4.  Rules. — These  should  also  be  deduced  and  stated  by  the 
pupils.     The  true  order  is  this  :  1.  A  mastery  of  the  process 
without  reference  to  the  rule.     2.  The  recognition  of  the  suc- 
cessive steps  in  order,  and   the  statement  of  each.     3.  The 
combination  of  these  several  statements  into  a  general  state- 
ment.    4.  A  comparison  of  this  generalization  with  the  au- 
thor's rule.     5.   The    memorizing  of  the  latter.     Take   for 
illustration  the  rule   for  adding  fractions.      T.  What  is  the 
first  step?     "Write   the  fractions,  separating  them  by  the 
plus  sign."     (Pupils   write  an  example.)     T.  What  is  the 
second  step?     "Keduce  the  fractions  to  a  common  denomi- 
nator."    T.  What  is  the  third  step?     "Add  the  numerators 
of  the  new  fractions."     T.   The  fourth  step?    "Under  their 
sum   write   the    common     denominator."      These   questions 
should  be  repeated  until  the  answers  are  promptly  and  ac- 
curately given,  and  then  they  should  be  united  in  a  general 
statement.     The  first  step  may  be  omitted  in  the  rule. 

5.  Questions  for  Review. — These  are  designed  as  a  final  test 
of  the  pupil's  knowledge.     Before  they  are  reached,  the  defi- 
nitions, principles,  and  rules  should  be  thoroughly  mastered, 
and  the  pupils  should  be  able  to  make  a  topical  analysis  of 
them  and  recite  each  in  order. 

6.  Properties  of  Numbers. — But  one  method  of  finding  the 
greatest  common  divisor  and  least  common  multiple  is  given, 
namely,  by  factoring.     The  other  methods  are  of  no  practical 
use  to  pupils  of  this  grade,  and  this  is  introduced  mainly  to 
give  practice  in  factoring  numbers. 

7.  Fractions. — This  section  presents  only  the  elements  of 
Fractions,  and  these  in  the  simplest  manner.     The  subject 
will  be  more  exhaustively  treated  in  the  Complete  Arith- 
metic.    The  reduction  of  compound  fractions  is  made  intro- 
ductory to  the  multiplication  of  fractions,  as  the  two  pro- 
cesses are  best  taught  together. 


SECTION   I. — NOTATION  AND  NUMERATION. 

PAGK 

Oral  and  Written  Exercises 9 

Definitions,  Principles,  and  Rules 16 

Roman  Notation 22 

Questions  for  Review 23 

SECTION  II.— ADDITION. 

Mental  and  Written  Exercises 24 

Definitions,  Principles,  and  Rule 38 

SECTION  III.— SUBTRACTION. 

Mental  and  Written  Exercises 40 

Definitions,  Principles,  and  Rule 49 

SECTION  IV.— MULTIPLICATION. 

Mental  and  Written  Exercises     .......  53 

Definitions,  Principles,  and  Rule .65 

SECTION  V.— DIVISION. 

Mental  and  Written  Exercises 70 

Definitions,  Principles,  and  Rule         ......  80 

Questions  for  Review  of  Simple  Rules 86 

SECTION  VI.— PROPERTIES  OF  NUMBERS. 

Divisor  and  Factor 87 

Greatest  Common  Divisor    ........  90 

Multiple  and  Least  Common  Multiple         .         .         .         .         .91 

SECTION  VII.— COMMON  FRACTIONS. 

The  Idea  of  a  Fraction  developed — Definition   ....  94 

Reduction  of  Integers  and  Mixed  Numbers  to  Fractions  .         .  97 

Reduction  of  Fractions  to  Integers  or  Mixed  Numbers      .         .  99 

Reduction  of  Fractions  to  their  Lowest  Terms   ....  '101 

Reduction  of  Fractions  to  Higher  Terms 103 

Reduction  of  Fractions  to  a  Common  Denominator    .         .         .  104 

Addition  of  Fractions 105 

Subtraction  of  Fractions 107 

Multiplication  of  Fractions  by  Integers Ill 

(vii) 


viii  CONTENTS. 

PA  G  K 

Fractional  Parts  of  Integers 112 

Multiplication  of  Integers  by  Fractions 114 

Reduction  of  Compound  Fractions       .         .          .          .         .         .115 

Multiplication  of  Fractions  by  Fractions    .         .          .         .         .117 

Division  of  Fractions  by  Integers 117 

Division  of  Integers  by  Fractions 119 

Division  of  Fractions  by  Fractions 120 

Numbers  Fractional  Parts  of  Other  Numbers      ....  121 

Questions  for  Review 124 

SECTION  VIII.— UNITED  STATES  MONEY. 

Notation  and  Definitions     .         .         .         .         .         .         .         .125 

Reduction 128 

Addition  and  Subtraction 130 

Multiplication  and  Division         .......  132 

Bills 137 

Questions  for  Review 141 

SECTION  IX. — REDUCTION  OF  DENOMINATE  NUMBERS. 

Dry  Measure 142 

Liquid  Measure    ..........  146 

Long  Measure      ..........  149 

Square  Measure 152 

Cubic  Measure 156 

Wood  Measure      ..........  158 

Circular  Measure           .         . 159 

Time  Measure 161 

Avoirdupois  Weight     .         .         .         .         .         .         .         .         .164 

Troy  Weight 167 

Apothecaries  Weight 169 

Miscellaneous  Table              ........  170 

Definitions,  Principles,  and  Rules 171 

Questions  for  Review            ........  174 

SECTION  X.— COMPOUND  NUMBERS. 

Compound  Addition — Problems 178 

Definitions,  Principles,  and  Rule  .  .  .  .  .  180 

Compound  Subtraction — Problems 181 

Definition  and  Rule 184 

Compound  Multiplication — Problems 184 

Definition  and  Rule 186 

Compound  Division — Problems 187 

Definition  and  Rules 189 

Miscellaneous  Problems 190 

Questions  for  Review 192 


*  ^ 

17  E! 


INTERMEDIATE  ARITHMETIC, 


SECTION     I. 

NOTATION 


LESSON  I. 

ORAL    EXERCISES. 

Article  1,  —  1.  Here  are  one  hundred  balls  in 
ten  rows.  How  many  balls  are  there  in  one  row? 
How  many  balls  in  two  rows?  In  three  rows? 
In  five  rows  ?  In  eight  rows  ?  In  ten  rows  ? 

(9) 


10  INTERMEDIATE    ARITHMETIC. 

2.  How  many  ones  in  ten?     How  many  ones  in 
two  tens  ?     In  five  tens  ?     Eight  tens  ?     Ten  tens  ? 

3.  How  many  tens  in  ten  ?     How  many  tens  in 
twenty?     In    thirty?      Forty?      Sixty?      Severity? 
Eighty  ?     One  hundred  ? 

Art,  2.  When  a  number  is  expressed  by  two  fig- 
ures, the  first  or  right-hand  figure  denotes  Units, 
and  the  second  or  left-hand  figure  denotes  Tens. 

4.  Which   figure    in    25    denotes    units  ?     Which 
denotes  tens? 

5.  How  many  tens   and  units    are   there   in  37  ? 
In    57?   46?   33?   50?   45?  64?    88?    94?    99? 

Art.  3.  In  reading  numbers,  the  tens  and  units 
are  read  together  as  so  many  units.  Thus,  45  is 
read  forty-five  units,  or,  more  briefly,  forty-five. 

Read  the  following  numbers,  and  give  the  num- 
ber of  tens  and  units  in  each : 


(6) 

(*) 

(8) 

(9) 

(10) 

(11) 

1 

11 

21 

20 

14 

67 

3 

13 

23 

40 

34 

83 

5 

15 

25 

60 

•  55 

75 

9 

19 

29 

80 

95 

72 

Art.  4.  When  a  number  is  expressed  by  three 
figures,  the  third  or  left-hand  figure  denotes  Hun- 
dreds. 

12.  Which    figure    in     245     denotes    hundreds? 
Which  figure  denotes  tens  ?     Which  denotes  units  ? 

13.  How  many  hundreds,  tens,  and  units  in  426  ? 
708?    340?    235?    406?    560?    666? 


NOTATION     AND    NUMERATION.  11 

Read  the  following  numbers,  and  give  tho  num- 
ber of  hundreds,  tens,  and  units  in  each : 


(14) 

(15) 

(16) 

(17) 

(18) 

200 

240 

302 

349 

560 

500 

550 

805 

424 

703 

700 

770 

307 

825 

909 

900 

990 

804 

448 

836 

19.  What   is    the    greatest   number   that   can   be 
expressed    by    one    figure  ?     By    two    figures  ?     By 
three  figures  ? 

20.  When  numbers  are   expressed  by  figures,  in 
which  place    or  order  is   the   figure   denoting  units 
written  ?     The    figure    denoting    tens  ?     The    figure 
denoting  hundreds  ? 

Art.  5.    The   first  three  figures,  viz. :   units,  tens, 
and  hundreds,  constitute  the  first  or  Units'  Period. 

WRITTEN    EXERCISES. 

1.  Write  in  words,  4,    6,    8,    13,    14,    18,    20, 
24,    30,    34. 

2.  Write  in  words,  40,     46,     60,    67,    70,    78, 
80,    83,    87,    90,    95,  99. 

Express  in  figures  the  following  numbers  : 


(3) 

(4) 

(5) 

Twelve  ; 

Twenty-one  ; 

Twenty-three  ; 

Sixteen; 

Thirty-two  ; 

Twenty-four  ; 

Eighteen  ; 

Forty-two  ; 

Forty-seven  ; 

Twenty  ; 

Sixty-five  ; 

Sixty-five  ; 

Sixty; 

Eighty-five; 

Seventy-nine  ; 

Eighty. 

Ninety-four. 

Ninety-six. 

12  INTERMEDIATE    ARITHMETIC. 

6.  Express  in  figures  the  numbers  composed  of 
three,  tens  and  four  units ;  six  tens  and  seven  units ; 
seven  tens  and  six  units  ;   seven  tens. 

7.  Express  in  figures  the  numbers  composed  of 
six  tens  and  eight  units ;  three  tens  and  nine  units ; 
nine  tens  and  no  units ;  seven  units. 

8.  Write  in  words,   100,   150,   200,    280,    300, 
350,  390,  560,  607,  803,  340,  and  908. 

Express  in  figures  the  following  numbers : 

(9)  (10) 

Two  hundred  ;  Four  hundred  and  five ; 

Five  hundred ;  Five  hundred  and  six  ; 

Seven  hundred;  Six  hundred  and  four; 

Three  hundred  and  forty;  Four  hundred  and  forty-five; 

Six  hundred  and  seventy ;  Eight  hundred  and  thirty-seven ; 

Nine  hundred  and  thirty.  Nine  hundred  and  twenty-seven. 

11.  Express  in  figures  the  numbers  composed  of 
three  hundreds,  five  tens,  and  four  units ;   six  hun- 
dreds,  four  tens,    and  three   units ;    five   hundreds, 
seven  tens,  and  no  units. 

12.  Express  in  figures  the  numbers  composed  of 
eight  hundreds  and  six  tens ;  five  hundreds  and  four 
tens ;   seven  hundreds  and  five  units ;   two  hundreds 
and  six  units ;    six  tens. 

13.  What   number   is    composed    of  3    hundreds, 
0  tens,  and  6  units?     2  hundreds  and  3  tens?     4 
hundreds  and  6  units  ?     5  hundreds  and  8  tens  ? 

14.  What   number   is    composed    of   5    tens    and 
8  units?     6   hundreds   and    5    units?     7  hundreds 
and  6  tens  ? 


NOTATION    AND    NUMERATION.  13 

LESSON     II. 
ORAL    EXERCISES. 

Thousands'   'Period— Thousands,  Ten- thousands, 
Hun  dred-  th  o  us  an  ds . 

Art.  6.  When  a  number  is  expressed  by  four 
figures,  the  fourth  or  left-hand  figure  denotes 
Thousands. 

1.  How    many    thousands     in    4,635?      3,045? 
6,309?     7,554?     5,384?     8,054?     5,006  ? 

2.  Read    the    units'    period    in    6,325  ;     5,080 ; 
7,009;    3,406;    5,800;    6,370;    7,590;    8,008. 

Read  the  following  numbers : 


(3) 

(4) 

(5) 

(6) 

(7) 

1,000 

2,200 

1,020 

2,007 

3,432 

3,000 

4,400 

3,040 

4,001 

4,568 

5,000 

6,600 

5,060 

5,003 

5,608 

7,000 

8,800 

7,090 

6,005 

7,893 

9,000 

9,900 

9,070 

8,009 

9,890 

Art.  7.  When  a  number  is  expressed  by  five 
figures,  the  fifth  or  left-hand  figure  denotes  tens 
of  thousands,  or  Ten-thousands. 

8.  How  many  ten-thousands  in  45,684  ?   50,480  ? 
38,305?   15,056?    80,650? 

9.  How   many   ten-thousands    and   thousands   in 
36,308?   48,500?    60,070?    85,350?    90,308? 

Art.  8.  In  reading  a  number  expressed  by  five 
figures,  the  fifth  and  fourth  figures  are  read  to- 
gether as  so  many  thousands.  Thus,  45,000  is 
read  forty-five  thousand. 


14  INTERMEDIATE    ARITHMETIC. 

Read  the  following  numbers: 

(10)  (11)  (12)                       (13) 

10,000  •           21,000  34,400  53,333 

30,000  44,000  53,440  16,089 

50,000  63,000  67,444  99,008 

70,000  84,000  48,307  28,045 

90,000  99,000  39,600  67,909 

Art,  9.  When  a  number  is  expressed  by  six  fig- 
ures, the  sixth  or  left-hand  figure  denotes  hundreds 
of  thousands,  or  Hundred-thousands. 

14.  How   many    hundred-thousands   in   534,000? 
308,000?    650,430?    508,080? 

15.  How  many  hundred-thousands,  ten-thousands, 
and    thousands    in    354,000?     607,800?     350,307? 
193,240?    470,386? 

Art.  10.  In  reading  a  number  expressed  by  six 
figures,  the  sixth,  fifth,  and  fourth  figures  are  read 
together  as  thousands.  Thus,  452,000  is  read  four 
hundred  and  fifty -tivo  thousand. 

Read  the  following  numbers  : 


(16) 

(17) 

(18) 

(19) 

200,000 

250,000 

845,630 

603,408 

400,000 

360,000 

803,084 

490,732 

600,000 

580,000 

760,432 

308,400 

800,000 

730,000 

900,425 

600,550 

900,000  960,000  807,708  707,700 

Art.  11.  The  fourth,  fifth,  and  sixth  figures  of  a 
number  constitute  the  Thousands'  Period. 

20.  Read  the  thousands'  period  in  the  16th, 
17th,  18th,  and  19th  examples. 


NOTATION     AND    NUMERATION.  15 

21.  How  many  orders  in  units'  period?  In 
thousands'  period? 

•22.  What  are  the  names  of  the  three  orders  in 
units'  period  ?  In  thousands'  period  ? 

23.  How  may  the  two  periods  be  separated  ? 
Ans.  By  a  comma. 

WRITTEN    EXERCISES. 

1.  Write  in  words,  3000;   4060;    3580;    7086; 
6606;   and   8080. 

2.  Write  in  words,  4400;    5008;    6070;    8506; 
5087;   7600;   and   3003. 

3.  Express    in    figures,    three    thousand ;    seven 
thousand ;    nine  thousand ;    four  thousand  five  hun- 
dred ;   eight  thousand  nine  hundred. 

4.  Express   in   figures,  two  thousand  four  hun- 
dred  and    forty ;    four   thousand    six    hundred   and 
sixty ;   five  thousand   eight  hundred ;    six  thousand 
five  hundred  and  twenty-five. 

5.  Express   in    figures,   seventy-five ;    two    hun- 
dred and  forty;   three  hundred  and  six;    five  hun- 
dred and  forty-five  ;    four  thousand. 

6.  Express  in   figures  four  hundred  and  forty ; 
five    hundred    and    ninety ;    seven    thousand    eight 
hundred ;    eight  thousand  and  fifty. 

7.  Write    in    words,    10000;     25000;     40500; 
36000;   44000;   30400;   45080;    64008;    89800. 

8.  Express   in    figures,   forty-five   thousand  five 
hundred   and  four ;    sixty   thousand    seven   hundred 
and    ninety ;     thirty-eight    thousand     and    twenty ; 
ninety-six  thousand  and  eighty-four. 


16  INTERMEDIATE     ARITHMETIC. 

9.  Express  in  figures,  four  hundred  and  twenty; 
seven  hundred  and  eighty-nine  ;  four  thousand  and 
fifty-seven ;  seventy-five  thousand ;  sixteen  thou- 
sand and  ninety-eight. 

10.  Express  in  figures,  as  one  number,  87  thou- 
sand 327  units ;   60  thousand  405  units ;    70  thou- 
sand 346  units ;    4  thousand  40   units ;   5  thousand 
5  units ;    95  thousand  406  units. 

11.  Express  in  figures,  as  one  number,  88  thou- 
sand 88   units ;    8  thousand  80  units ;    65  thousand 
60  units ;    6  thousand  600  units ;    60  thousand. 

12.  Write  in  words,  300000;    440000;   334000; 
245500;    304800;   450340. 

13.  Express  in  figures,  four  hundred  thousand; 
six    hundred    thousand ;    eight    hundred    and    forty 

%  thousand ;   seven  hundred  and  sixty  thousand. 

14.  Express  in    figures,    nine    hundred   and  fifty 
thousand  four  hundred ;    four  hundred  and  fifty-five 
thousand  two  hundred  and  eighty. 

15.  Separate  the  following  numbers  into  periods  : 
3080;   44004;   400080;   20066;   109038;   160006; 
809090;  706030;   40004;    30030. 

LESSON  III. 
DEFINITIONS,  PEINOIPLES,  AND  EULES, 

Art,  12.    Arithmetic   is  the    science   of  num- 
bers, and  the  art  of  numerical  computation. 

A  Number  is  a  unit  or  a  collection   of  units. 
A  Unit  is  one  thing  of  any  kind. 
An  Integer  is  a  whole  number. 


NOTATION     AND     NUMERATION.  17 

Art.  13.  There  are  three  methods  of  expressing 
numbers : 

1.  By  words;   as,  five,  fifty,  etc. 

2.  By  letters,  called  the  Roman  method.  (Art.  23.) 

3.  By  figures,  called  the  Arabic  method. 

Art.  14.  Notation  is  the  art  of  expressing 
numbers  by  figures  or  letters. 

Numeration  is  the  art  of  reading  numbers 
expressed  by  figures  or  letters. 

The  word  Notation  is  commonly  used  to  denote  the  Arabic 
method,  which  expresses  numbers  by  figures. 

Art.  15.  In  expressing  numbers  by  figures,  ten 
characters  are  used,  viz. :  0, 1,  2,  3,  4,  5,  6,  7,  8,  9. 

The  first  of  these  characters,  0,  is  called  Naught, 
or  Cipher.  It  denotes  nothing,  or  the  absence  of 
number. 

The  other  nine  characters  are  called  Significant 
Figures,  or  Numeral  Figures.  They  each  ex- 
press one  or  more  units.  They  are  also  called 
Digits. 

Art.  16.  The  successive  figures  which  express  a 
number,  denote  successive  Orders  of  Units.  These 
orders  are  numbered  from  the  right;  as,  first,  sec- 
ond, third,  fourth,  fifth,  and  so  on. 

A  figure  in  units'  place  denotes  units  of  the  first 
order;  in  tens'  place,  units  of  the  second  order;  in 
hundreds'  place,  units  of  the  third  order,  and  so 
on  —  the. term  units  being  used  to  express  ones  of 
any  order, 

I.-  A.— 2. 


18  INTERMEDIATE    ARITHMETIC. 

Art.  17.  Ten  units  make  one  ten,  ten  tens  make 
one  hundred,  ten  hundreds  make  one  thousand ; 
and,  generally^  ten  units  of  any  order  make  one 
unit  of  the  next  higher  order. 

NOTE. — The  teacher  can  make  this  principle  plain  by 
means  of  the  illustration  given  on  page  9.  It  is  easily  shown 
that  10  ones  or  units  equal  1  ten,  and  that  10  tens  equal 
1  hundred. 

Art.  18.  Figures  have  two  values,  called  Simple 
and  Local. 

The  Simple  Value  of  a  figure  is  its  value  when 
standing  alone. 

The  Local  Value  of  a  figure  is  its  value  arising 
from  the  order  in  which  it  stands. 

When  3,  for  example,  stands  alone,  or  in  the 
first  order,  it  denotes  3  units;  when  it  stands  in 
the  second  order,  as  in  34,  it  denotes  3  tens;  when 
it  stands  in  the  third  order,  as  in  354,  it  denotes 
3  hundreds.  Hence,  the  local  value  of  figures  in- 
creases from  right  to  left  in  a  tenfold  ratio. 

The  local  value  of  each  of  the  successive  figures 
which  express  a  number,  is  called  a  Term.  The 
terms  of  325  are  3  hundreds,  2  tens,  and  5  units. 

Art.  19.  The  figures  denoting  the  successive  or- 
ders of  units,  are  divided  into  groups  of  three 
figures  each,  called  Periods.  The  first  or  right- 
hand  period  is  called  Units;  the  second,  Thou- 
sands ;  the  third,  Millions ;  the  fourth,  Billions ; 
the  fifth,  Trillions;  the  sixth,  Quadrillions;  the 
seventh,  Quintillions,  etc. 


NOTATION    AND    NUMERATION.  19 

Art.  20.  The  three  orders  of  any  period,  count- 
ing from  the  right,  denote,  respectively,  Units, 
Tens,  and  Hundreds,  as  shown  in  the  table: 


"     " 

fi        •••* 
£         fl 


555,444,333,222,111 

5th  Period,     4th  Period,     3d  Period,     2d  Period,      1st  Period, 
Trillions.        Billions.       Millions.      Thousands.        Units. 

The  several  orders  may  be  named  more  briefly 
by  calling  the  first  order  of  each  period  by  the 
name  of  the  period,  and  omitting  the  word  "  of" 
after  tens  and  hundreds,  thus  : 


3       TS 

2     fl 


555,444,333,222,111 

5th  Period.     4th  Period.     3d  Period.     2d  Period.      1st  Period. . 

Art.  21.  RULE  FOR  NOTATION. — Begin  at  the  left, 
and  write  the  figures  of  each  period  in  their  proper 
orders,  filling  all  vacant  orders  and  periods  with 
ciphers. 


20  INTERMEDIATE    ARITHMETIC. 

Art.  22.  RULE  FOR  NUMERATION.  —  1.  Begin  at 
the  right,  and  separate  the  number  into  periods  of 
three  figures  each. 

2.  Begin  at  the  left,  and  read  each  period  con- 
taining one  or  more  significant  figures  as  if  it  stood 
alone,  adding  its  name. 

NOTE.  —  The  name   of  the  units'  period  is  usually  omitted. 

WRITTEN    EXERCISES. 

1.  Write  in  words,  20080406. 

SUGGESTION.  —  Separate  the  number  into  periods,  thus: 
20,080,406.  Then  write  each  period,  thus:  Twenty  million 
eighty  thousand  four  hundred  and  six. 

2.  Write  in  words,  50038456. 

3.  Write  in  words,  300607008. 

4.  Write  in  words,  40000300400. 

SUGGESTION.  —  Omit  the  third  period,  since  it  contains  no 
significant  figures,  thus:  Forty  billion  three  hundred  thousand 
four  hundred. 

5.  Write  in  words,  3450000067. 

6.  Read  3000080040;   10080603400. 

7.  Read  15000407030;    5075803004. 

8.  Read  400440300500 ;   130030003003. 

9.  Express    in    figures,  twelve    billion    forty-six 
million  and  nine. 

PROCESS.  —  First,  write  12,  with  a  comma  after  it,  to 
form  the  fourth  or  billions'  period,  thus :  12, ;  then  write 
46  in  the  next  period,  filling  the  vacant  order  with  a 
cipher,  thus :  12,046, ;  then,  as  there  are  no  thousands, 
fill  the  next  three  orders  with  ciphers,  thus:  12,046,000,; 
and,  finally,  write  9  in  the  units'  period,  filling  the  vacant 
orders  with  ciphers,  thus:  12,046,000,009. 


NOTATION    AND    NUMERATION.  21 

10.  Express   in   figures,    fifty   million    thirty-two 
thousand  six  hundred  and  forty. 

11.  Three    hundred    million    nine    thousand    two 
hundred  and  six. 

12.  Forty-eight   billion   seventeen  thousand  and 
sixty-four. 

13.  Five  million  five  thousand  and  five. 

14.  One   million  one  hundred  thousand   and  ten. 

15.  Three   trillion   three    hundred    million    three 
hundred  and  three. 

16.  Sixty-two    million    three    hundred    thousand 
and  forty-nine. 

17.  Five  hundred  million  five  thousand. 

18.  Four   hundred   and    six    thousand   five   hun- 
dred and  seven. 

19.  Two  million  ten  thousand  and  eighty. 

20.  Ninety  million  seven  thousand  four  hundred 
and  ninety. 

21.  Four    hundred    million    forty   thousand   four 
hundred  and  four. 

22.  Thirty  billion  seventy-five  thousand. 

23.  Nine  billion  nine  thousand  and  nine. 

24.  Fifty-four  million  eighty-seven  thousand  and 
eighty-six. 

25.  Two   hundred    and    two   thousand  five   hun- 
dred  and   eighty. 

26.  Fifty  billion  fifty  million  five  hundred  thou- 
sand and  seven. 

27.  Seventeen    billion    seven    hundred    thousand 
three  hundred  and  six.     ' 

28.  Ninety  million  ten  thousand  and  fifty-five. 


22  INTERMEDIATE    ARITHMETIC. 

LESSON  VII. 


Art.  23.  In  the  Roman  Notation,  numbers  are 
expressed  by  means  of  seven  capital  letters,  viz.  : 
I,  V,  X,  L,  C,  D,  M. 

I  stands  for  one  ;  V  for  five  ;  X  for  ten  ;  L  for 
fifty  ;  C  for  one  hundred  ;  D  for  five  hundred  ; 
M  for  one  thousand. 

Art.  24.  All  other  numbers  are  expressed  by  re- 
peating or  combining  these  letters. 

1.  When   a   letter  is   repeated,   its   value   is   re- 
peated ;    thus:    II    represent    2;    XX,    20;    COG, 
300,  etc. 

2.  When  a  letter  is  placed  before  one  of  greater 
value,   the   less   value   is   taken   from   the   greater; 
thus:    IV  stands  for  4;    IX  for  9;    XC  for  90. 

3.  When  a  letter  is  placed  after  one  of  greater 
value,  the  less  value  is  added  to  the  greater,  thus  : 
VI  stands  for  6  ;   XI  for  11  ;    CX  for  110. 

Art.  25.  In  the  following  table,  numbers  are  ex- 
pressed by  letters  and  figures  : 


I, 

1; 

VIII, 

8; 

XV, 

15; 

XL, 

40; 

II, 

2; 

IX, 

9; 

XVI, 

16; 

L, 

50; 

III, 

3; 

x, 

10; 

XVII, 

17; 

LX, 

60; 

IV, 

4: 

XI, 

11; 

XVIII, 

18; 

LXX, 

70; 

v, 

5; 

XII, 

12; 

XIX, 

19; 

LXXX, 

80; 

VI, 

6; 

XIII, 

13; 

XX, 

20; 

XC, 

90; 

VII,  7;  XIV,  14;  XXX,    30;  C,          100. 


NOTATION    AND    NUMERATION.  23 

WRITTEN    EXERCISES. 

Express  the  following  numbers  in  figures : 

(1)  (2)                                     (3) 

XIV  COL                       MDCL 

XXIV  DCXC                   MDLX 

XXXIX  CCXG                    MDLIX 

XCVI  DCCL                    MDCCC 

CXI  DCLIX                 MDCCCLX 

CIX  MCCL                   MDCCCLXIX 

Express  the  following  numbers  by  letters : 

(4)  (5)                           (6)                            (7) 

45  150                      210                      1500 

76  184                     550                      1650 

1)0  345                      700                      1850 

93  433                      750                      1868 

99  555                      880                      1940 

Express  the  following  numbers  by  letters: 

(8)  (9)                             (10)                           (11) 

204  1200                     1685                     2000 

409  1350                     1944                     2050 

540  1408                     1865                     2550 

675  1590                     1909                    3010 


QUESTIONS  FOR  REVIEW. 

What  is  arithmetic?  What  is  a  number?  A  unit? 
An  integer? 

In  how  many  ways  may  numbers  be  expressed  ?  How 
are  numbers  expressed  in  the  Arabic  method?  In  the 
Roman  method  ?  What  is  notation  ?  What  is  numera- 
tion ? 


24  INTERMEDIATE    ARITHMETIC. 

How  many  figures  are  used  to  express  numbers? 
Which  are  called  significant  figures?  Which  has  no 
numerical  value  ? 

What  is  meant  by  orders  of  units?  How  are  the  or- 
ders numbered  ?  How  many  units  of  any  order  make  one 
unit  of  the  next  higher  order? 

What  is  meant  by  the  simple  value  of  a  figure?  On 
what  does  the  local  value  of  a  figure  depend?  What  is 
the  law  of  increase  from  right  to  left  ? 

How  many  orders  make  a  period  ?  What  are  the 
names  of  these  orders  ?  Give  the  names  of  the  first  six 
periods.  Give  the  rule  for  notation.  Give  the  rule  for 
numeration. 

How  are  numbers  expressed  in  the  Roman  notation  ? 
Name  the  letters  used,  and  give  the  value  of  each.  How 
are  numbers  expressed  by  these  letters? 


SECTION     II. 

A'D'DITION. 


LESSON   I. 
A.ddltire  Numbers,   /,  2,  and  3. 

1.  Four   and    2    are   how   many?     8  and    2?     6 
and  2?     7  and  2  ?     3  and  2?     9  and  2  ? 

2.  Two    and   3    are    how   many?     5    and    3?     6 
and  3  ?     7  and  3  ?     9  and  3  ?     11  and  3  ? 

3.  How   many   are   8    and    3?     18    and  3?     38 
and  3?     47  and  3?     67  and  3?     87  and  3? 

4.  How    many    are  9    and    2  ?     39   and    2  ?     59 
and  2  ?     48  and  3  ?     38  and  3  ?     88  and  3  ? 


ADDITION.  25 

5.  Frank   has    5    marbles   in    one    hand    and    2 
marbles   in   the   other :    how  many  marbles  has   he 
in  both  hands  ? 

SOLUTION.  —  5    marbles    and    2    marbles    are    7    marbles: 
Frank  has  7  marbles  in  both  hands? 

6.  A  drover  bought  9  sheep  of  one  farmer  and 
2  sheep  of  another :    how  many  sheep  did  he  buy  ? 

7.  Jane    spelled   17    words    correctly    and   mis- 
spelled 3  :    how  many  words   did  she  try  to   spell  ? 

8.  A  grocer  sold  8  pounds  of  sugar  to  one  cus- 
tomer, 3  pounds  to  another,  and  2  pounds   to   an- 
other :    how  many  pounds  of  sugar  did  he  sell  ? 

9.  A  man  walked  4  miles  the  first  hour,  3  miles 
the  second,  and  2  miles  the  third :    how  many  miles 
did  he  walk  in  the  three  hours  ? 

10.  Begin  with  1  and  count  to  45  by  adding  2 
successively,  thus:    1,  3,  5,  7,  9,  11,  13,  etc. 

11.  Begin  with  2  and  count  to  50  by  adding  3 
successively. 

NOTE. — This  drill  should  be  continued  until  the  class  can 
add  by  2's  or  3's  with  rapidity  and  accuracy. 

WRITTEN    EXERCISES. 

Add  the  following  numbers : 

(1)       (2)        (3)        (4)          (5) 

12102 
21210 
10222 
11121 
21212 
12111 


2 

10 

112 

2112 

o 

21 

211 

1201 

1 

12 

122 

1122 

2 

20 

111 

2021 

1 

22 

222 

1212 

2 

11 

121 

2221 

26  -  INTERMEDIATE    ARITHMETIC. 

(6)  Write   the    numbers    so    that   the   .units 

PHOCESS.  shall  form  the  first  column;   the  tens,  the 

12i  second    column;    and    the    hundreds,   the 

233  third  column.     Begin  with  the   units'  col- 

123  umn,  and  add,  naming  results  only,  thus: 

332  3,  5,   8,  11,    12,   14,  17,   20,    21  —  21   units 

231  equal  2  tens  and  1  unit.     Write  the  1  unit 

under  the   units'   column,   and    add  the   2 
tens  with  the  tens'  column,  thus :   5,  8,  10, 
^  12,  15,  18,  20,  23,  25—25  tens  equal  2  hun- 

. — -  dreds  and  5  tens.     Write  the  5  tens  under 

2051,  Sum.  the  tens>  column,  and  add  the  2  hundreds 
with  the  hundreds'  column,  thus :  5,  7,  8, 
11,  13,  16,  17,  19,  20  —  20  hundreds  equal  2  thousands  and 
0  hundreds.  Write  the  0  hundreds  under  the  hundreds' 
column,  and  write  the  2  thousands  in  thousands'  place. 
The  sum  is  2  thousands,  0  hundreds,  5  tens,  and  1  unit, 
or  2051.  To  test  the  accuracy  of  the  work,  add  the  col- 
umns downward. 

(T)  (8)  (9)  (10) 

13  232  1323  3232 

22  123  2112  2323 

20  212  2131  23213 

31  131  3213  13221 
12  120  1301  32233 

21  102  2222  232111 

23  223  1111  323212 

32  121  1323  232021 

11.  Add  213,    322,    203,    312,    222,    321,    231, 
123,   303,   232,  311,   132. 

12.  What  is  the  sum  of  2132,  3113,  2323,  1313, 
2132,  and  3320? 

13.  2021  + 12333  +  22031  +  332231  +  231323  - 
how  many? 


ADDITION.  27 

14.  3231  +  2302  +  2330  +  12332  ===  how  many  ? 

15.  A  grocer  sold  12   pounds    of  sugar   to    one 
customer,    21    pounds    to    another,     32    pounds    to 
another,   and    30    pounds    to    another :    how   many 
pounds  did  he  sell  ? 

16.  July  has  31  days ;    August,  31 ;    September, 
30 ;    October,   31 ;   November,    30 ;    and   December, 
31 :    how   many    days    in    the   last   six   months    of 
the   year  ? 

17.  A   farm    contains    120    acres,    another    212 
acres,  another  133   acres,  and   another  322  acres  : 
how  many  acres  do  the  four  farms  contain  ? 

18.  A  man    bought   four  loads  of  hay,  the  first 
weighing   2130   pounds,   the    second   2312    pounds, 
the  third  2232  pounds,  and  the  fourth  2322  pounds : 
how  many  pounds  of  hay  in  the  four  loads  ? 

LESSON  II. 

MENTAL    EXERCISES. 
JVew  JLdditire  Numbers,  £.   and  5. 

1.  Three    and  4    are    how   many?     5    and    4? 
6  and  4?     8  and  4?     7  and  4?     9  and  4? 

2.  Two  and  5    are  how  many  ?     4  and   5  ?     6 
and  5?     8  and  5?     7  and  5?     9  and  5? 

3.  How  many  are  18  and  4?     28  and  4?     48 
and  4?     16  and  4?     36  and  4  ?     56  and  4? 

4.  How  many  are  17  and  5?     27  and  5?     47 
and  5?     29  and  5  ?     49  and  5?     69  and  5  ? 

5.  There    are   17  birds    on    one   tree   and  4  on 
another :   how  many  birds  on  both  trees  ? 


28  INTERMEDIATE    ARITHMETIC. 

6.  A  man   gave    26    dollars    for   a    coat   and   5 
dollars   for  a   hat :    how   many  dollars   did   he   give 
for  both  ? 

7.  A  drover  bought  19   cows  of  one  man   and 
4  of  another :    how  many  cows  did  he  buy  ? 

8.  James    picked    27    peaches    from    one    limb 
and  5    peaches   from   another :    how  many   peaches 
did  he  pick  from  both  limbs  ? 

9.  Mary  has   written   16   lines :   if  she  write   5 
lines   more,    how   many    lines    will    she    then    have 
written  ? 

10.  George    gave    15    cents    for   a    slate    and   5 
cents  for  a  pencil :    how  many  cents   did  he   give 
for  both? 

11.  Begin  with  2  and  count  to  50  by  adding  4 
successively. 

12.  Begin  with  3  and  count  to  48  by  adding  5 
successively. 

WRITTEN    EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(*) 

15 

251 

15215 

23512 

52134 

25 

153 

14343 

30425 

34445 

35 

354 

45046 

41341 

53054 

45 

452 

50350 

23301 

44052 

55 

355 

33432 

41545 

25253 

45 

254 

43543 

43453 

34545 

35 

555 

23343 

25445 

41534 

25 

444 

45452 

41505 

22335 

6.  What  is  the  sum  of  four  hundred  and  four ; 
four  thousand  and  forty ;  forty  thousand  four  hun- 
dred;  and  four  million  four  hundred  thousand? 


ADDITION.  29 

7.  A  grain  dealer  bought  2350  bushels  of  wheat 
on  Monday,  4215  bushels  on  Tuesday,  3245  bushels 
on  Wednesday,   1500   bushels    on   Thursday,   2424 
bushels  on  Friday,  and  1350  bushels  on  Saturday : 
how  many  bushels  did  he  buy  ? 

8.  In   a    city  containing   five   wards,  there  are 
345   voters   in   the   first  ward,  443  in  the   second, 
213  in  the  third,  523  in  the  fourth,  and  425  in  the 
fifth :   how  many  voters  in  the  city  ? 

9.  A  father  gave  to   his   eldest  son  225   acres 
of  land,  to  the  second  155  acres,  to  the  third  145 
acres,  and  to  the  youngest  124    acres :    how  many 
acres  did  he  give  to  all  ? 

10.  The  first  three  cars  of  a  freight  train  contain 
35240  pounds  each ;  the  next  four  cars,  25345 
pounds  each ;  the  next  two  cars,  31540  pounds 
each ;  and  the  last  car,  25432  pounds  :  how  many 
pounds  of  freight  in  the  ten  cars  ? 

LESSON   III. 

MENTAL    EXERCISES. 
New  dLdditive  Number,  6. 

1.  Two   and  6  are  how  many?     4   and   6?     3 
and  6?     5  and  6?     7  and  6  ?     9  and  6?     8  and  6? 

2.  How  many  are  17  and  6?     28  and  6  ?     48 
and  6?     68  and  6?     58  and  6?     78  and  6  ? 

3.  How  many  are  19   and   6  ?     29  and  6  ?     59 
and  6?     39  and  6?     69  and  6  ?     49  and  6? 

4.  Begin  with  3  and  count  to  63  by  adding  6 
successively. 


30          INTERMEDIATE:  ARITHMETIC. 

5.  Mary's  father  gave   her  5   peaches   and   her 
mother    gave    her  6 :    how  many  peaches   did  both 
give  her  ? 

6.  John   solved  18   problems  before   school  and 
6  problems  in  school:    how  many  problems  did  he 
solve  in  all  ? 

7.  A  farmer  bought   a   cow  for  27  dollars   and 
a   calf  for   6    dollars :    how    many    dollars    did    he 
pay  for  both  ? 

8.  The  head  of  a  fish  is  5  inches  long,  its  body 
16    inches,    and    its    tail    6    inches :     how    long    is 
the  fish? 

9.  In    a    certain    orchard    there    are    29    apple 
trees,  5  pear  trees,  and  6  peach  trees :    how  many 
trees  in  the  orchard? 

10.  William  gave  a  blind  boy  19  cents,  John 
gave  him  15  cents,  and  Charles  6  cents:  how 
many  cents  did  they  all  give  him? 

WRITTEN     EXERCISES. 


(t) 

(2) 

(3) 

(4) 

(5) 

3640 

24137 

43260 

35260 

305129 

2566 

16126 

32345 

16165 

224603 

1654 

20050 

16606 

32542 

350164 

2366 

16654 

46060 

36344 

255234 

3456 

33456 

50050 

24030 

145344 

5634 

44162 

16566 

33246 

242456 

4565 

23206 

24656 

21438 

145346 

5656 

36562 

32562 

44546 

200500 

6.    Add  thirty-six   thousand    three   hundred    and 
twenty-five ;      fourteen     thousand     and     forty-six ; 


ADDITION.  31 


twenty-three  thousand  four  hundred  and  five; 
fifteen  thousand  and  sixteen ;  and  three  hundred 
and  six  thousand  three  hundred  and  four. 

7.  What  is  the  sum  of  three  million  one  thou- 
sand and  fifty-six ;  six  hundred  thousand  six  hun- 
dred and  twenty-five  ;  four  million  forty-two  thou- 
sand and  four ;  forty-five  million  six  hundred  and 
fifty  thousand? 

LESSON    IV. 

MENTAL     EXERCISES. 
New  Additive  Number,  f. 

1.  Two    and    7    are    how   many?     5    and    7?     3 
and  7  ?     6  and  7  ?     8  and  7  ?     7  and  7  ?     9  and  7? 

2.  How   many  are  18    arid   7?     48    and   7?     68 
and  7?     88  and  7?     28  and  7? 

3.  Fifteen    and    7    are   how   many  ?     35  and   7  ? 
65  and  7?     45  and  7?     75  and  7  ? 

4.  Begin  with  4   and   count  to   53  by  adding  7 
successively. 

5.  Charles   had^B   marbles   and   his    father  gave 
him  7  :    how  many  marbles  had  he  then  ? 

6.  A  garden  contains  19  pear  trees  and  7  peach 
trees :    how  many  trees  in  the  garden  ? 

7.  A  man  bought  a   set  of  harness   for  37  dol- 
lars and  a  saddle  for  7  dollars  :    how  much  did  he 
pay  for  both? 

8.  Mr.  Jones  gave  8  plums  to  John,  6  to  Henry, 
and  7  to  George:    how  many  plums  did  he  give  to 
the  three  boys  ? 


32  INTERMEDIATE    ARITHMETIC. 

9.  Frank  gave  10  cents  for  a  lead-pencil,  5 
cents  for  a  piece  of  rubber,  and  7  cents  for  paper : 
how  much  did  the  three  articles  cost? 

10.  A  gentleman  gave  36  dollars  for  a  suit  of 
clothes,  7  dollars  for  a  pair  of  boots,  and  5  dol- 
lars for  a  hat:  how  much  did  he  pay  for  all? 

WRITTEN    EXERCISES. 

(1)  (2)                        (3)                          (4) 

10640  24045  32620  7121365 

14075  14036  75437  2171634 

26507  25507  50743  1237773 

16021  46364  64017  7143656 

34412  54563  32516  2674467 

53452  16057  18416  6734765 

26123  72027  13673  6574636 

16021  47735  31654  7147347 


5.  What   is    the    sum    of   sixteen    million    four 
thousand   and   sixty-five ;    three    hundred   thousand 
twb    hundred    and    fifty-six ;    seven    thousand    and 
forty ;    and  five  million  five  thousand  and  seven  ? 

6.  What  is  the  sum  of  fortf%ive  million   seven 
thousand   and  seventy ;    six  million  sixty-five  thou- 
sand two  hundred  and  six ;    and  seventy -five  thou- 
sand and  forty -four  ? 

7.  January  has   31    days ;   February  (except  in 
leap  year),   28 ;    March,  31 ;    April,  30 ;    May,  31 ; 
and   June    30 :    how   many    days    in    the    first    six 
months  of  the  year? 

8.  A   gentleman    owns    five    farms,   containing, 
respectively,   285  acres,  345  acres,  146  acres,  438 


ADDITION.  33 

acres,  and   248    acres  :    how   many   acres    of   land 
does  he  own  ? 

9.  A  newsboy  sold  327  papers  in  April,  465 
in  May,  318  in  June,  and  278  in  July  :  how  many 
papers  did  he  sell  in  the  four  months? 

10.  The  first  ward  of  a  city  contains  1675  youth 
of  school  age ;  the  second,  2357  youth ;  the  third, 
2347 ;  the  fourth,  3270 ;  and  the  fifth,  2677 :  how 
many  youth  of  school  age  in  the  city  ? 

LESSON   V. 

MENTAL    EXERCISES. 
New  Additive  Number,  8. 

1.  Two   and   8    are   how  many?     5  and   8?     3 
and  8  ?     6  and  8  ?     4  and  8  ?     9  and  8  ? 

2.  How  many  is   16  plus   8?     36  plus  8?     56 
plus   8?     25  plus  8?     45  plus  8?     65  plus  8? 

3.  How   many   is   13  +  8?     33  +  8?     53  + 
8?     29  +  8?     49  +  8?     69  +  8? 

4.  Begin  with  3  and   count  to  51  by  adding  8 
successively. 

5.  Jane    solved    17    problems    in    the    morning 
and    8    in    the    evening :    how   many   problems    did 
she  solve  ? 

6.  A  farmer  raised  16  loads    of  wheat  in   one 
field  and  8  loads  in  another :    how  much  wheat  did 
he   raise  ? 

7.  Kate    spelled    38   words    correctly   and   mis- 
spelled 8 :    how  many  words   did  she  try  to   spell  ? 

8.  Charles  gave  25   cents    for   a   speller  and  8 

I.  A.— 3. 


34  INTERMEDIATE    ARITHMETIC. 

cents    for   a    pencil :    how    much    did    he    give   for 
both? 

9.  A  lady  paid  27  dollars  for  a  shawl,  8  dol- 
lars for  a  bonnet,  and  3  dollars  for  a  pair  of 
shoes :  how  much  did  she  pay  for  all  ? 

10.  A  merchant  sold  18  yards  of  muslin  to  one 
customer,  7  yards  to  another,  and  8  yards  to  an- 
other :  how  many  yards  did  he  sell  ? 

•WRITTEN    EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(5) 

308 

2617 

19864 

42764 

5868 

280 

4565 

34687 

38768 

4384 

667 

6387 

46768 

34187 

5065 

444 

7836 

65837 

63506 

6008 

555 

5666 

80040 

24483 

4873 

371 

4084 

18608 

43832 

8345 

736 

8168 

36084 

41608 

6654 

644 

7846 

45687 

37860 

5636 

6.  Add   thirty  thousand  six  hundred  and  fifty ; 
fifty  thousand  and  eighty-five ;    four  hundred  thou- 
sand  six   hundred    and   seven ;    arid   three    hundred 
and  forty  thousand  and  seventy. 

7.  Add  eight  million  eight  thousand  and  eight ; 
eighteen   million    eighteen    thousand   and    eighteen ; 
and  eight  hundred  million  eight   hundred  thousand 
eight  hundred. 

8.  The   distance  by  railroad   from   Philadelphia 
to    Harrisburg   is    106    miles;    from    Harrisburg  to 
Pittsburg,  249  miles ;   from  Pittsburg  to   Crestline, 
188    miles;    from    Crestline    to    Fort   Wayne,   132 


ADDITION.  35 

miles ;    from  Fort  Wayne   to    Chicago,  148   miles : 
how  far  is  it  from  Philadelphia  to  Chicago  ? 

9.  One  of  the  wards  of  a  certain  city  contains 
1384  houses ;  another,  2868  houses ;  another,  857 
houses ;  and  another,  1486  houses :  how  many 
houses  in  the  city  ? 

10.  A  steamship  sailed  217  miles  the  first  day; 
265  miles  the  second;  227  miles  the  third;  187 
miles  the  fourth;  and  168  miles  the  fifth:  how 
many  miles  did  it  sail  in  the  five  days? 

LESSON    VI. 

MENTAL    EXERCISES. 
New  JLdditire  Number,  9. 

1.  Three  and  9  are  how  many  ?     7  and  9  ?     9 
and  7?     8  and  9?     9  and  8?     5  and  9  ? 

2.  How  many  is  14  +  9  ?     24  +  9  ?     44  +  9  ? 
16  +  9?     36  +  9?     56  +  9? 

3.  How  many  is  17  +  9  ?     37  +  9  ?     57  +  9  ? 
23  +  9  ?     43  +  9  ?     63  +  9  ? 

4.  Begin  with  3  and  count  to  57  by  adding  9 
successively. 

5.  A  farmer  sold    6    hogs   to   his   neighbor  and 
9  to  a  drover :    how  many  hogs  did  he  sell? 

6.  Andrew  sold  8  bunches   of  grapes  and  had 
9  bunches  left :  how  many  bunches  had  he  at  first  ? 

7.  There  are  17  cows  in  one  field  and  9  cows 
in  another :    how  many  cows  in  both  fields  ? 

8.  A  pole   is   7   feet  in   the  water  and    9  feet 
in  the  air:    how  long  is  the  pole? 


36  INTERMEDIATE    ARITHMETIC. 

9.  A  man  paid  23  dollars  for  a  coat,  9  dol- 
lars for  a  pair  of  pants,  and  8  dollars  for  a  vest : 
how  much  did  he  pay  for  the  suit  ? 

10.  A  boy  paid  45  cents  for  a  ball,  8  cents  for 
marbles,  and  7  cents  for  an  orange :  how  much 
did  he  pay  for  all£- 

WKITTEN    EXERCISES. 


(1) 

(2) 

(3) 

(4) 

(5) 

57384 

4369 

45566 

48 

4868 

5834 

13846 

806 

76 

3769 

691 

3482 

9376 

287 

1804 

2637 

691 

2038 

80 

786 

13484 

5873 

4056 

409 

5863 

596 

578 

8705 

96 

4836 

43486 

509 

6508 

378 

3988 

6.  What  is  the  sum   of  nine  billion   nine   mill- 
ion and  nine ;    nine   hundred  million  nine  hundred 
thousand   nine    hundred ;    and    ninety    million    nine 
hundred  thousand  and  ninety  ? 

7.  The    State  of  Maine  contains  31766  square 
miles ;    New  Hampshire,  9280    square  miles ;    Ver- 
mont,   10212 ;    Massachusetts,    7800 ;    Connecticut, 
4674  ;   and  Rhode  Island,  1306.     How  many  square 
miles  in  all  of  the  New  England  States  ? 

8.  The    distance    by    railroad    from    Boston    to 
Springfield   is   98   miles ;    from    Springfield  to   Al- 
bany,   103    miles ;    from    Albany    to    Buffalo,    298 
miles;    from  Buffalo  to  Cleveland,  183  miles;    from 
Cleveland   to    Chicago,    355    miles.     How   far  from 
Boston  to  Chicago? 


ADDITION.  37 

LESSON  VIII. 


1.  An  orchard  contains   25  apple   trees    and  8 
peach  trees  :   how  many  trees  in  the  orchard  ? 

2.  A  gardener    sold   17  quarts   of  strawberries 
in  market   and  9   quarts   to   a   grocer  ;    how  many 
quarts  did  he  sell  ? 

3.  A   lady   gave   15    cents   for  thread,  8  cents 
for  needles,  and  7  cents  for  pins  :   how  many  cents 
did  she  spend? 

4.  James  gave  8  cherries  to  George,  7  to  Wil- 
liam, 6  to  Thomas,  9  to  Harry,  and  kept  5  :    how 
many  cherries  had  he  at  first? 

5.  A  gentleman   gave    95   dollars  for   a   horse, 
15  dollars  for  a  saddle,  and  5  dollars  for  a  bridle  : 
how  much  did  he  pay  for  all  ? 

6.  Begin  with  2  and   add   to   72   by  7's,  thus  : 
9,  16,  23,  30,  37,  etc. 

7.  Begin  with  5  and  add  to  61  by  8's. 

8.  Begin  with  3  and  add  to  69  by  6's. 

9.  Begin  with  4  and  add  to  67  by  9's. 

WKITTEN    EXERCISES. 

1.  32545  +  8607  +  11709  +  50063  =  how  many  ? 

2.  A  man   paid   $3575    for  a  lot,  $5450  for  a 
house,   $875  for  a   stable,   and  $675  for  other  im- 
provements :   what  did  the  property  cost  him  ? 

NOTE.  —  This  character   ($)    denotes   dollars,   and  is  called 
the  dollar  sign:  $35  is  read  35  dollars;  $1  is  read  1  dollar. 


38  INTERMEDIATE    ARITHMETIC. 

3.  The    first    book    of    a     series     contains    328 
pages ;     the    second,    392    pages ;    the    third,   400 
pages;     and    the    fourth,    432    pages:     how    many 
pages  in  the  series  ? 

4.  Ohio  contains  39964  square  miles ;   Michigan, 
56243  square  miles ;   Indiana,  33809  square  miles ; 
and  Illinois,  55409  square  miles:   what  is  the  area 
of  these  four  States  ? 

5.  The    distance    by   railroad   from   Pittsburg   to 
Columbus  is  193  miles ;   from  Columbus  to  Cincin- 
nati, 120  miles  ;    from  Cincinnati  to  St.  Louis,  340 
miles :    how  far  is  it  from  Pittsburg  to   St.  Louis  ? 

6.  A  father  divided  his  estate  between  two  sons 
and    three    daughters,   giving    to    each    son    $3250 
and  to  each  daughter  $2750 :  what  was  the  value 
of  the  estate  ? 

7.  A   farmer    raised    in    one    year    380    bushels 
of  wheat,  245   bushels  of  oats,  87  bushels  of  rye, 
and  as   many  bushels   of   corn   as    of  wheat,  oats, 
and  rye  together:    how  many  bushels  of  grain  did 
he  raise? 

DEFINITIONS,  PRINCIPLES,  AND  RULE, 

Art.  26.  Addition  is  the  process  of  finding 
the  sum  of  two  or  more  numbers. 

The  number  obtained  by  adding  two  or  more 
numbers  is  called  the  Sum  or  Amount. 

The  Sum  contains  as  many  units  as  all  the 
numbers  added,  taken  together. 


ADDITION.  39 

Numbers   are   either   Concrete   or  Abstract. 

A  Concrete  Number  is  applied  to  a  par- 
ticular thing  or  quantity;  as,  4  pears,  7  hours, 
30  steps. 

An  Abstract  Number  is  not  applied  to  any 
particular  thing  or  quantity;  as,  4,  7,  30. 

Fourteen  balls  and  13  balls  are  numbers  of  the  same 
kind;  and  6  tens  and  3  tens  are  numbers  of  the  same 
order.  Numbers  of  the  same  kind  or  order  are  called 
Like  Numbers.  Only  like  numbers  can  be  added. 

Art.  27.  The  Sign  of  Addition  is  + .  It  is  called 
plus  meaning  more.  When  placed  between  two 
numbers,  it  shows  that  they  are  to  be  added. 
Thus,  8  +  5  is  read  8  plus  5,  and  it  shows  that  5 
is  to  be  added  to  8. 

The  Sign  of  Equality  is  — .  It  is  read  equals 
or  is  equal  to.  Thus  7  +  8  =  15  is  read  7  plus  8 
equals  15. 

Art.  28.  RULE  FOR  ADDITION. — 1.  Write  the  num- 
bers to  be  added  so  that  figures  denoting  units  of  the 
same  order  shall  be  in  the  same  column,  and  draw 
a  line  underneath. 

2.  Beginning  with  units,  add  each  column,  and 
write  the  sum,  when  less  than  ten,  underneath. 

3.  When   the    sum    of  any   column  exceeds   nine, 
write  the  right-hand  figure  under  the  column  added, 
and  add  the  number  denoted  by  the  left-hand  fig- 
ure or  figures  with  the  next  column. 

4.  Write  the  entire  sum  of  the  left-hand  column. 
PROOF. — Add  the  columns  doivnward. 


40  INTERMEDIATE    ARITHMETIC. 


SEOTIOIV     III. 


LESSON   I. 
Subtrahend  Figures,    /,  2,  3. 

1.  How  many  is  4  less  3?     6  less  3?     8  less  3? 
10  less  3?     12  less  3?     11  less  3? 

2.  How  many  is  11  less  2?     21  less  2?     41  less 
2?     19  less  2?     29  less  2?     49  less  2? 

3.  Three    from   12    leaves    how   many  ?     3   from 
22?     3   from   42?     3   from    52?     3    from    32?     3 
from  20?     3  from  40?     3  from  50? 

4.  Begin    with    50    and     count    back    to    0    by 
subtracting  2   successively,  thus :    50,    48,   46,  44, 
42,  etc. 

5.  Begin    with    40    and    count    back    to    1    by 
subtracting   3    successively. 

6.  Charles   bought   12    sticks   of  candy  and  ate 
3   of  them :    how  many   sticks   were  left  ? 

7.  The  teacher  pronounced  21  words  to  Henry, 
and   he    misspelled   2    of  them :    how   many   words 
did  he  spell  correctly  ? 

8.  A   lesson    in    arithmetic    consists    of   15    ex- 
amples, and  Charles  has  solved  all  but  3  of  them: 
how  many  examples  has  he  solved  ? 

9.  James     is    11     years     old    and    his    brother 
Henry  is  3  years  younger :   how  old  is  Henry  ? 


SUBTRACTION.  41 

"WRITTEN    EXAMPLES. 

1.   From  345  take  123. 
PROCESS.  Write  123  under  345,  placing  units 

Minuend,       345          Under    Units>     tens    Under  *tens>    and 

Subtrahend    123         hundreds  under  hundreds.     Subtract  3 

units  from  5  units,  and  write  2  units, 

Difference,      222          the  difference>  below;    subtract  2  tens 

from  4  tens,  and  write  2  tens,  the 
difference,  below ;  subtract  1  hundred  from  3  hundreds, 
and  write  2  hundreds,  the  difference  below.  The  difference, 
or  remainder,  is  222. 

(2)  (3)  (4)  (5)  (6)  (7)  (8) 

57  46  88  75  685  409  967 

43  24  65  53  343  307  645 

(9)  (10)  (11)  (12)  (13)  (14)  (15) 

246  487  507  718  563  485  560 

132  231  302  312  330  2^2  320 

16.  From  four  thousand  and  sixty-five  take  two 
thousand  and  thirty-one. 

17.  A  grocer  bought    585  pounds  of  sugar  and 
sold  231  pounds:    how  many  pounds  had  he  left? 

18.  In  a  graded  school,  there  are  345  boys  and 
321  girls :    how  many  more  boys  than  girls  in  the 
school  ? 

LESSON   II. 

MENTAL    EXEKCISES. 
JV°en>  Subtrahend  Figures,  £  and  5. 

1.   How  many  is  7  less   4?      6  less  4?     9  less 
4?     8  less  4?     10  less  4?     11  less  4? 


42  INTERMEDIATE     ARITHMETIC. 

2.  How  many  is  7  less   5  ?     9  less  5  ?     8  less 
5?     10  less  5?     12  less  5?     15  less  5? 

3.  How  many  is  13   less   4?     23   less   4?     43 
less    4?      63    less    4?      83    less    4?      53    less    4? 
93  less  4? 

4.  How  many  is  14   less   5  ?     44  less   5  ?     34 
less  5  ?     54  less  5  ?     64  less   5  ?     74  less  5  ?     94 
less  5  ? 

5.  Begin    with    60    and    count    back    to    0    by 
subtracting   4   successively. 

6.  Begin    with    53    and    count    back    to    1    by 
subtracting   4   successively. 

7.  A  man  gave  $12  for  a  saddle  and  $4  for  a 
bridle  :    how  much  did  the   saddle   cost  more   than 
the  bridle? 

8.  Charles   earned   21    cents   by  selling  papers 
and    gave   4   cents   for   a   comb :    how  many    cents 
had  he  left? 

9.  Kate   is    15    years   old   and   her    sister  is   4 
years  younger  :    what  is  her  sister's  age  ? 

10.  There  are  21  passengers  in  a  car :    if  5  of 
them  leave  at  a  station,  how  many  will  remain? 

11.  There  are  13  men  in  one  coach  and  5  men 
in    another :    how    many    men    in    the    first    coach 
more  than  in  the  second? 

WRITTEN    EXERCISES. 

(1)        (2)        (3)        (4)       (5)         (6) 
335     2036      308      1565      3683     7863 
214     1034     205      1433      2542      4552 


SUBTRACTION.  43 

7.  From    five    thousand    and    seventy-six    take 
three  thousand  and  fifty. 

8.  A  farm  contains  358  acres  of  land:   if  155 
acres  should  be  sold,  how  many  would  be  left? 

9.  A  man  bought  a  house  for  $4320  and  sold 
it  for  $6450 :   how  much  did  he  gain  ? 

10.  A  man  bought   3487   bushels   of  wheat  and 
sold  1425  bushels:   how  many  bushels  had  he  left? 

11.  A    ship-builder   sold    a    vessel    for   $24350 : 
if    the    vessel    cost    him    $27585,    how    much    did 
he  lose? 

12.  The    number    of    school-houses    in    Ohio,    in 
1867,   was   11353;    in  Pennsylvania,   11453:    how 
many    more    school-houses    in    Pennsylvania    than 
in   Ohio? 

13.  A  wool  dealer  having  bought   23437  fleeces 
of   wool,    shipped    12322   fleeces    to    Boston:    how 
many  fleeces  had  he  left? 

LESSON   III. 

MENTAL    EXERCISES. 
New   Subtrahend  Figures,    6  and   7 . 

1.  How   many  is    8   less    6?  10    less    6?     12 
less  6?      13  less  6?     15  less  6?  14  less  6? 

2.  How  many  is  12  less   7?  13    less    7?     15 
less  7?     16  less  7?     18  less  7?  22  less   7? 

3.  How  many  is  14  less   6?  24    less    6?     44 
less  6  ?     64  less  6  ?     34  less  6  ?  74  less  6  ? 

4.  How  many  is   14   less  7  ?  24  less  7  ?     44 
less  7?     16  less  7?     36  less  7?  46  less  7? 


44  INTERMEDIATE    ARITHMETIC. 

5.  Begin    with    56    and    count    back    to    0    by 
subtracting  7  successively. 

6.  Begin    with    60    and    count    back    to    0    by 
subtracting  6  successively. 

7.  Ella   was    absent    from    school    7   days    in   a 
term  of  75  days  :  how  many  days  was  she  .present  ? 

8.  John    earned    25    cents    by   selling    oranges 
and   gave    6    cents    for    a   lead-pencil :    how   many 
cents  had  he  left? 

9.  A  boy  was  carrying  home  21  eggs ;    he  fell 
and  broke  7  of  them  :    how  many  were  left  ? 

10.  A   man    having    23    dollars,    gave    6   dollars 
for  a  hat :    how  many  dollars  had  he  left  ? 

11.  A  teacher  pronounced  25  words   to   an  idle 
pupil,  who  misspelled  7  of  them :   how  many  words 
did  he  spell  correctly? 

WRITTEN    EXERCISES. 

1.   From  5334  take  2726. 

PROCESS.  Since  6  units  can  not  be  taken  from  4 

Min     5334          units,  add  10  units  to  the  4  units,  making 

Sub  '   2796          -^  un^s  >    then   subtract  6   units  from   14 

'  units,    and    write    8   units,    the    difference, 

below.      To  balance  the  10  units  (equal  1 

ten)   added  to  the   minuend,  add  1  ten  to 

the  2  tens  of  the  subtrahend;   then   subtract  3  tens  from 

3  tens,  and  write  0  tens,  the  difference,  below. 

Add  10  hundreds  to  the  3  hundreds  of  the  minuend, 
making  13  hundreds;  subtract  7  hundreds  from  13  hun- 
dreds, and  write  6  hundreds,  the  difference,  below.  To 
balance  the  10  hundreds  (equal  1  thousand)  added  to  the 
minuend,  add  1  thousand  to  the  2  thousands  of  the  sub- 


SUBTRACTION.  45 

trahend ;   subtract  3  thousands  from  5  thousands,  and  write 

2  thousands,  the  difference,  below.     The  difference  is  2608. 

This  process  may  be  shortened,  thus :  6  units  from  4 
units  plus  10  units,  or  14  units,  leave  8  units;  2  tens  and 
1  ten  are  3  tens,  and  3  tens  from  3  tens  leave  0  ten; 
7  hundreds  from  3  hundreds  plus  10  hundreds,  or  13  hun- 
dreds, leave  6  hundreds;  1  thousand  and  2  thousands  are 

3  thousands,   and   3   thousands  from   5  thousands  leave  2 
thousands.     The  difference  is  2608. 

NOTE.  —  The  teacher  should  show  that  the  adding  of  10  to 
a  term  of  the  minuend  and  1  to  the  next  higher  term  of  the 
subtrahend,  increases  both  minuend  and  subtrahend  equally, 
and  does  not  affect  the  difference. 

(2)  (3)  (4)  (5)  (6) 

44  63  272  1385  5754 

26  46  147  1276  3457 

(7)  (8)  (9)  (10)  (11) 

3416  3041  14406  20670  30401 

2507  2637  7345  17356  20576 

12.  From  fourteen  thousand  and  forty-four  take 
six   thousand  and  sixteen. 

13.  A  man  whose  income  is  $1850  expends  an- 
nually $1365:   how  much  does  he  lay  up? 

14.  The   number   of    youth    of   school   age   in   a 
certain    city    is    1234,    and    only    756    pupils    are 
enrolled  in  the    schools  :   how  many  youth   do   not 
attend  school  ? 

15.  The  number  of  pupils   enrolled   in  the  pub- 
lic schools  of  Ohio,  in  1867,  was  704767 ;   in  Penn- 
sylvania,   789389 :    how    many    more    pupils    were 
enrolled  in  Pennsylvania  than  in  Ohio  ? 


46  INTERMEDIATE    ARITHMETIC. 

LESSON   IV. 

MENTAL    EXERCISES. 
New  Subtrahend  Figures,  8  and  ,9. 

1.  How   many   is    9    less    8?     11    less    8?     13 
less  8  ?     10  less  8  ?     14  less   8  ?     12  less  8  ? 

2.  How  many  is   16   less  8?     26   less  8?     56 
less  8?     17  less  8?     27  less  8?     67  less  8? 

3.  How  many  is   11   less    9?     13  less   9?     15 
less  9?     12  less  9?     16  less  9?     17  less  9? 

4.  How  many  is  16    less   9?     26  less   9?     36 
less  9?     15  less  9?     25  less  9?     45. less  9? 

5.  Begin    with    50    and    count    back    to    2    by 
subtracting  8  successively. 

6.  Begin    with    57    and    count    back    to    3    by 
subtracting  9  successively. 

7.  A  school  has   enrolled  65  pupils,  and  8  are 
absent :    how  many  are  present  ? 

8.  Mr.    Smith    is    44    years    of    age    and    his 
youngest  son  is    8  years    of  age  :   what  is  the  dif- 
ference in  their  ages  ? 

9.  A  school   contains  9   more   girls  than  boys : 
if  there  are  56  girls,  what  is  the  number  of  boys  ? 

WRITTEN    EXERCISES. 

1.  From  800000  take  238. 

2.  From  forty  million  take  eighty  thousand. 

3.  A    nursery    contains    705    peach    trees    and 
428    plum    trees :     how    many    more    peach    trees 
than  plum  trees  in  it? 


SUBTRACTION.  47 

4.  The   Pilgrims   landed   at  Plymouth  in  1620, 
and    our    National    Independence    was    declared    in 
1776:   how  many  years  between  the  two  events? 

5.  The  first    steamboat  was  made  in  1807,  and 
the   Atlantic    Cable  was   laid   in   1866  :    how  many 
years  between  the  two  events? 

6.  Mont  Blanc   in  Europe   is  15668   feet   high, 
and    Mount    Sorata    in    South    America    is    21286 
feet    high  :    what    is    the    difference    is   the    height 
of  these  two  mountains  ? 

7.  A  man  who    owned    3408    sheep,   sold  1897 
of  them  :    how  many  sheep  had  he  left  ? 

8.  Mt.  Etna  is  10874  feet  high,  and  Mt.  Vesu- 
vius  3948    feet  :    how    much    higher   is    Etna   than 
Vesuvius  ? 

9.  America   was   discovered   in   1492,    and    the 
Pilgrims  landed  at  Plymouth  in  1620  :    how  many 
years  intervened  ? 

10.  The  population  of  the  State  of  New  York 
in  1860  was  3881000,  and  that  of  Ohio,  2340000  : 
how  many  more  people  in  New  York  than  in 
Ohio? 

LESSON    V. 


1.  From     a     cask     containing     45     gallons     of 
molasses,  39  gallons  were  sold  :   how  many  gallons 
remained  unsold  ? 

2.  An    orchard    contains    56    apple    trees    and 
48  peach  trees  :   how  many  more   apple  trees  than 
peach  trees  in  the  orchard  ? 


48  INTERMEDIATE    ARITHMETIC. 

3.  A  grocer  sold   57  pounds   of  butter  from  a 
firkin    containing    65    pounds :    how   many   pounds 
remained  in  the  firkin  ? 

4.  In  a  school,  63   pupils  are  enrolled  and  54 
are  present:    how  many  pupils  are  absent? 

5.  If  a   man   earns    $45    a  month,   and    spends 
$36:    how  much  does  he  lay  up? 

6.  A  man  gave  $75   for  a  watch    and   $22  for 
a  chain  :    how  much  did  the  watch  cost  more  than 
the  chain  ? 

7.  Charles   has   17    marbles    and   John   8 :    how 
many  more  marbles  has  Charles  than  John? 

8.  A  teacher  asked  his  class   52  questions,  and 
8    were    answered    incorrectly :     how    many    were 
answered  correctly  ? 

9.  In    a    term    of   64    days,    Charles    attended 
school  55  days :   how  many  days  was  he  absent  ? 

10.  Subtract   by   4's  from   62   back  to   2,  thus: 
62,  58,  54,  50,  46,  etc. 

11.  Subtract  by  6's  from  75  back  to  3. 

12.  Subtract  by  9's  from  68  back  to  5. 

13.  Subtract  by  7's  from  59  back  to  3. 

14.  Subtract  by  8's  from  48  back  to  0. 

"WRITTEN    EXERCISES. 

1.  From  202380  take  165436. 

2.  4308560  —  1674805  =  how  many  ? 

3.  Illinois    contains    55409    square    miles,    and 
Missouri    67380    square    miles :    how    much    more 
area  has  Missouri  than  Illinois? 


SUBTRACTION.  49 

4.  By  the  census  of  1850  the  entire  population 
of  the  United   States    was    23191876,   and    by   the 
census   of   1860    it   was    31224885:    what   was    the 
increase  in  10  years  ? 

5.  In  1862  there  were  10869   miles   of  railroad 
in   Great   Britain,  and    33222   miles   in  the  United 
States :   how  many  more  miles  in  the  United  States 
than  in  Great  Britain  ? 

6.  An  army  of  30340  men  lost  7568  in  battle : 
how  many  men  did  it  then  contain  ? 

7.  In  1862  Ohio  produced  35442858  pounds  of 
butter  and  20637235  pounds  of  cheese :    how  much 
more  butter  than  cheese  was  produced  ? 

8.  A  merchant   having   $11315   in   bank,    drew 

out  $978  :    how  much  remained  in  the  bank  ? 

« 

DEFINITIONS,  PEINCIPLES,  AND  EULE, 

Art.  29.  Subtraction  is  the  process  of  find- 
ing the  difference  between  two  numbers. 

The  Difference  or  Remainder  is  the  num- 
ber found  by  subtracting  one  number  from  an- 
other. 

The  Minuend  is  the  number  diminished. 

The  Subtrahend  is  the  number  subtracted. 

Art.  30.  Only  Like  Numbers  can  be  subtracted. 
We  can  not  take  3  pencils  from  7  books,  nor  3 
units  from  7  tens. 

Art.  31.  The  Sign  of  Subtraction  is  — .  It  is 
read  minus  or  less.  It  shows  that  the  number  after 
it  is  to  be  subtracted  from  the  number  before  it. 

I.  A.— 4. 


50  INTERMEDIATE    ARITHMETIC. 

Art.  32,  RULE  FOR  SUBTRACTION.  —  1.  Write  the 
subtrahend  under  the  minuend,  placing  units  un- 
der units,  tens  under  tens,  hundreds  under  hun- 
dreds, etc. 

2.  Begin  at  the  right,  and  subtract  each  term  of 
the   subtrahend  from   the   term  above    it,    and   write 
the  difference  underneath. 

3.  When  any   term   of  the    subtrahend  is  greater 
than  the  term  above  it,   add  10   to  the   upper  term, 
and  then  subtract,  and  write  the  difference  as  before. 

4.  When  ten  has  been    added  to    the  upper  term, 
add   1    to    the   next   higher   term    of  the    subtrahend 
before  subtracting. 

PROOF.  —  Add  the  remainder  and  subtrahend. 
If  their  suJn  is  equal  to  the  minuend,  the  work 
is  correct. 

NOTE.  —  Instead  of  adding  1  to  the  next  term  of  the  sub- 
trahend, 1  may  be  subtracted  from  the  next  term  of  the 
minuend. 

LESSON  VI. 
"Problems  combining  Addition   and  Subtraction. 

1.  Robert  picked  21  peaches,  and  gave  7  to  his 
sister   and    8    to    his    brother :    how   many  peaches 
had  he  left? 

2.  A   garden    contains   17    pear    trees,    8    plum 
trees,    and    a    certain    number   of   peach    trees  :    if 
there    are    33    trees    in    the    garden,    what    is    the 
number  of  peach  trees  ? 

3.  A  grocer  bought  35   bushels   of  apples,   and 


SUBTRACTION.  51 

sold  17  bushels  to  A,  9  bushels  to  B,  and  the  rest 
to  C  :   how  many  bushels  did  he  sell  to  C  ? 

4.  Jane  is   8   years  old  and  Lucy  13,  and  the 
sum   of  Jane's  and  Lucy's   ages,   less    7   years,   is 
the  age  of  Mary :   how  old  is  Mary  ? 

5.  A  man  bought  a  firkin  of  butter  for  $17,  a 
crock  of  lard  for  $8,  and  a  barrel  of  flour  for  $9  ; 
but  he   had  not  money  enough   by   $7  to   pay   for 
them :    how  much  money  had  he  ? 

6.  A  man  earned  $45,  and  paid  $15  for  house 
rent,  $8  for  flour,  $7  for  shoes,  and  $10   for  gro- 
ceries :   how  much  had  he  left  ? 

7.  A  man  sees  15  pigeons  on  one  branch  of  a 
tree,  and  9  pigeons  on  another  branch :  if  7  should 
fly  away,  how  many  would  be  left  on  the  tree  ? 

8.  A  farmer  had  23    chickens,  but  7  of  them 
were   stolen   and    5    were    carried   off  by  a   hawk: 
how  many  chickens  had  he  left? 

9.  A  drover  bought  17   sheep   of  one    farmer, 
9  sheep    of   another,   and   8   of  another,   and   then 
sold  7   of   them  to    a   butcher :    how    many   sheep 
had  he  left? 

10.  A  man  gave  a  watch  and  $9  in  money  for 
a  horse  valued  at  $75 :  what  did  he  get  for  his 
watch  ? 

WRITTEN    EXERCISES. 

1.  From  a  piece  of  carpeting  containing  150 
yards,  a  merchant  sold  3  carpets,  containing  re- 
spectively 27,  39,  and  42  yards :  how  many  yards 
were  left  ? 


52  INTERMEDIATE    ARITHMETIC. 

2.  Rhode    Island     contains    an    area     of    1306 
square  miles  ;   Delaware,  2120  ;    Connecticut,  4674  ; 
New    Jersey,    8320;     Maryland,    9356;    and    New 
York,  47000 :    how  many   more    square    miles    has 
New  York  than  the  other  five  States  named? 

3.  A  regiment   entered   the   service    with  1088 
men :    150   were    killed    in    battle,    65    died    from 
disease,    24    deserted,    and    250    were    discharged : 
how  many  remained? 

4.  A  grain  dealer  bought  1250  bushels  of  wheat 
on  Monday,    2145   bushels   on   Tuesday,   and    3240 
bushels   on  Wednesday,   and    on   Thursday,   fearing 
a    decline    in    price,    he    sold    5450    bushels :    how 
much  wheat  had  he  left? 

5.  A  man    deposited    $175,    $141,   $75,   $304, 
and   $250    in    a    bank,    and    then    drew    out    $480 
and    $225  :     how    many    dollars    remained    in    the 
bank  ?  ^ 

6.  A  railroad  train  left  Cincinnati  for  St.  Louis 
with  336  passengers,  and  during  the  trip  145  pas- 
sengers   came    aboard,    and    208    passengers    left : 
how  many  were  in  the  cars  when  the  train  reached 
St.  Louis  ? 

7.  In     1860    the    population     of     Maine     was 
628279;      New     Hampshire,     326073;      Vermont, 
315098;      Massachusetts,     1231066;      Connecticut, 
460147;    Rhode   Island,  174620;    and   New  York, 
3880735.      How    many    more    inhabitants    in    New 
York  than  in  the  six  New  England  States  ? 

8.  A   man    gave    to   his  eldest   son   $2380 ;   to 
the  second,   $245  less   than   to   the  eldest ;   and.  to 


MULTIPLICATION.  53 

the  youngest,  $450    less  than  to  the   second :    how 
much  di.d  he  give  to  all? 

9.   From  the    sum  of  2348   and   1864  subtract 
their  difference. 

10.   From  the  sum  of  506703  and  340067  take 
their  difference. 


SEOTIO1V     IV. 

ATION. 


LESSON  I. 
Multiplicand  Figures,   1 ,  2,  and  3. 

1.  Twice    2    are    how    many  ?     4   times   2  ?     6 
times  2?     5  times  2?     8  times  2?     9  times  2? 

2.  Twice    3    are    how   many  ?     3   times    3  ?     5 
times  3  ?     4  times  3  ?     7  times  3  ?     9  times  3  ? 

3.  How  many  are  5  times  1  ?     5  times   3  ?     7 
times  1?     7  times  2?     8  times  1?     8  times  3? 

4.  A  boy  has  2  hands :   how  many  hands  have 
6  boys?     8  boys?     10  boys? 

5.  There  are  3  feet  in  a  yard :   how  many  feet 
in  2  yards  ?     4  yards  ?     6  yards  ? 

6.  If   3   bushels    of   apples    fill    a   barrel,    how 
many   bushels   will   fill   3   barrels?     5    barrels?     7 
barrels  ? 

7.  If  a  man   earn   3   dollars   a  day,  how  many 
dollars  will  he  earn  in  6  days  ? 


54  INTERMKDIATK    ARITHMKTIC. 

8.    If  a  boy  walk   3  miles   a  day  in   attending 
school,  how  many  miles  will   he  walk  in  10  days  ? 

WRITTEN    EXERCISES. 

1.   Multiply  232  by  3. 

Write  the  multiplier  3  under  the 


units'  figure  of  the  multiplicand, 
Multiplicand,  232  and  multipiy>  thus:  3  times  2  units 
Multiplier,  _  3  are  6  unitg  .  3  times  3  tens  are  9 

Product,  696          tens;  3  times  2  hundreds  are  6  hun- 

dreds.    The  product  is  696. 

(2)  (3)  (4)  (5)  (6) 

3212  10202  23321  202122  303203 

333  _  4  _  3 

7.  Multiply  230321  by  2.     By  3.  * 

8.  Multiply  320201  by  3.     By  4.     By  2. 

9.  If  a  gold  watch  is  worth  $220,  what  is  the 
worth  of  4  gold  watches  ? 

10.  A  drover  bought  3  horses   at   $133  apiece  : 
what  did  they  cost? 

11.  There   are   320   rods  in   a  mile  :   how  many 
rods  are  there  in  3  miles  ?     In  4  miles  ? 

LESSON    II. 

MENTAL    EXERCISES. 
New  Multiplicand  Figures,  &   and  5. 

1.  Twice  4  are  how  many?  3  times  4?  5 
times  4?  4  times  4?  6  times  4?  8  times  4? 
7  times  4?  9  times  4?  10  times  4? 


MULTIPLICATION.  55 

2.  Twice    5    are    how   many?     5    times    5?     6 
times  5?     8  times  5?     7  times  5?     9  times  5? 

3.  How    many    are    7    times    4  ?     7    times    5  ? 
9  times  4?     9  times  5?     10  times  5? 

4.  If  a  lemon  cost  4  cents,  what  will  6  lemons 
cost  ?     8  lemons  ?     10  lemons  ? 

5.  How   much  will   a   man    earn   in  7  days,  at 
$4  a  day  ?     In  8  days  ?     9  days  ? 

6.  There    are    5    cents    in    a    half-dime :    how 
many  cents  in  3  half-dimes  ?     5  half-dimes  ? 

7.  If  you  write  5  lines  a  day,  how  many  lines 
will  you  write  in  4  days  ? 

8.  If  5  boys  can  sit  on  one   bench,  how  many 
boys  can  sit  on  8  benches  ? 

9.  What   will    10    quarts    of    currants    cost,    at 
5  cents  a  quart  ? 

10.  If  there  are  5  school-days  each  week,  how 
many  school-days  are  there  in  6  weeks  ?  In  8 
weeks  ?  10  weeks  ? 


WRITTEN     EXERCISES. 

1.   Multiply  434  by  6. 

PROCESS.  Multiply  the  number  denoted  by 

each  figure  of  the   multiplicand  by 
Multiplicand.   434  ml  &       „    ,. 

f  6.     Thus  :    £   times   4   units  are  24 

units,  which  equal  2  tens  and  4  units; 


write  the  4  units  in  units'  place  in 
the  product,  and  reserve  the  2  tens. 

Six  times  3  tens  are  18  tens,  and  18  tens  plus  the  2  tens 
reserved  are  20  tens,  which  equal  2  hundreds  and  0  tens; 
write  the  0  tens  in  tens'  place  in  the  product,  and  reserve 


osr 


56  INTERMEDIATE    ARITHMETIC. 

the  2  hundreds.  Six  times  4  hundreds  are  24  hundreds, 
and  24  hundreds  plus  the  2  hundreds  reserved  are  25  hun- 
dreds, which  equal  2  thousands  and  6  hundreds;  write  the 
6  hundreds  in  hundreds'  place  in  the  product,  and  the 
2  thousands  in  thousands'  place.  The  product  is  2604. 

(2)  (3)  (4)  (5)  (6)  (7) 

453          2524          4545          3545          13545          25245 
86  7  8  4  6 


8.  If  there  are  324  pins  on  a  paper,  how  many 
pins  are  there  on  3  papers  ?     5  papers  ? 

9.  If  a  train  of  cars  run  425  miles  a  day,  how 
far  will  it  run  in  8  days  ? 

10.  If  135  tons  of  iron  rails  will  make  one  mile 
of  railroad,  how  many  tons  will  make  7  miles  ? 

11.  What  will  6  horses  cost,  at  $152  apiece? 

12.  A    father    divided    his    estate    between    four 
sons,  giving  to    each   $3545 :    what  was   the  value 
of  the  estate  ? 

13.  There    are    1440    minutes    in    a    day :    how 
many  minutes  in  7  days,   or  a  week  ? 

14.  If  it  take  15520   shingles  to  cover  a  house, 
how  many  shingles  will  cover  8  houses? 

LESSON   III. 

MENTAL     EXERCISES. 
New  Multiplicand  Figure,   6. 

1.  Twice  6  are  how  many?  4  times  6?  3 
times  6?  5  times  6?  7  times  6?  6  times  6? 
8  times  6?  9  times  6?  10  times  6?  .  . 


MULTIPLICATION.  57 

2.  There   are    8    rows    of   trees    in    an    orchard, 
and  6  trees  in  each  row :    how  many  trees   in  the 
orchard  ? 

3.  What   will    7    lead-pencils    cost,    at    6    cents 
apiece  ? 

4.  What  will  6  oranges  cost,  at  8  cents  apiece? 

5.  There    are    6    days   for   labor  in    each   week: 
how  many  days  for  labor  in  6  weeks  ?      9  weeks  ? 

6.  John    caught    6    fishes    and    Harry    7    times 
as  many  as  John:   how  many  did  Harry  catch? 

7.  If  a    horse   travel   6   miles   an   hour,  how  far 
will  it  travel  in  5  hours  ?     In  10  hours  ? 

8.  There   are    6    feet    in    a    fathom :    how   many 
feet  in  7  fathoms?     9  fathoms? 

•WRITTEN    EXERCISES. 

1.   Multiply  456  by  43. 

PROCESS.  Write  the  multiplier  under  the 

Multiplicand,       456          multiplicand,  placing  units  under 

Multiplier,              43          lmits  and  tens  under . tens'  .  First 
• multiply  by  the  3  units,  as  in  the 

Partial       j       1  preceding  lesson,  which  gives  1368 

cfs>  for  the  first  partial  product.     Next 

Product,          19608          multiply  by  the  4  tens,  observing 

that  units  multiplied  by  tens   (or 

tens  by  units)  produce  tens,  that  tens  by  tens  produce  hun- 
dreds, and  that  hundreds  by  tens  produce  thousands,  &c. 
This  gives,  for  the  second  partial  product,  4  tens,  2  hundreds, 
8  thousands,  and  1  ten-thousand,  which  are  to  be  written  in 
their  proper  orders,  since  unlike  orders  can  not  be  added. 
Then  add  the  two  partial  products,  and  their  sum,  which 
is  19608,  is  the  product  required. 


58  INTERMEDIATE    ARITHMETIC. 

NOTE.  —  The  teacher  should  show  that  units  multiplied  by 
tens  produce  tens;  tens  by  tens,  hundreds,  etc.  This  may  be 
done,  in  the  above  example,  by  changing  the  4  tens  into  40 
units.  40  times  6  units  =  240  units,  or  24  tens;  and  40  times 
5  tens  —  200  tens,  or  20  hundreds,  etc.  The  first  figure  of  each 
partial  product  is  written  under  the  multiplier  which  pro- 
duces it. 

(2)  (3)  (4)  (5)  (6)  (7)  (8) 

606        562        653         1446        2306        4636        40563 
54          67          86          234          726  67  143 


9.    If  a  ship  sail  216  miles  a  day,  how  far  will 
it  sail  in  38  days  ? 

10.  What  will  27  carriages  cost,  at  $165  apiece? 

11.  If  a  web   of  flannel  contain   46  yards,  how 
many  yards   in  897  webs  ? 

LESSON   IV. 

MENTAL     EXERCISES. 

New  Multiplicand  Figure,   ?. 

1.  Three  times  7  are  how  many?     5  times  7? 
7  times  7?     9  times  7?     8  times  7? 

2.  There    are    7   days    in   a   week :    how   many 
days  in  2  weeks  ?     4  weeks  ? 

3.  How  many  hills   of  potatoes  in  6  rows,  if 
there  are  7  hills  in  each  row  ? 

4.  If    Charles     earn    7     dollars    a    week,    how 
much  will  he  earn  in  5  weeks  ? 

5.  If  a   horse   travel  7  miles  in   an  hour,  how 
far  will  he  travel  in  8  hours  ? 

6.  If  5   men   can  build  a  wall  in  7  days,  how 
long  will  it  take  one  man  to  build  it? 


MULTIPLICATION.  59 

7.  If  a  box  of  crackers  will  last  8  men  7  days, 
how  long  will  it  last  one  man  ? 

8.  An  orchard  contains  10   rows   of  trees,  and 
there   are  7  trees    in   each"  row  :    how  many  trees 
in  the  orchard? 

WRITTEN    EXERCISES. 

1.   Multiply  2745  by  306. 

Multiply  successively  by  the  first 
and  third  figures  of  the  multiplier, 
observing  that  units  multiplied  by 
hundreds  produce  hundreds,  and 


Partial     j     16470        hence  writing  the  first  figure  of  the 
Products,  (8235  second  partial  product  in  hundreds' 

Product,      839970        ord«r-     In    306    there    are   no    tens 
to  be  used  as  a  multiplier. 

(2)  (3)  (4)  (5)  (6)  (7) 

4086         32607         7908          8099         60772         86507 
4008  4009  909          1088  1019  9003 

8.  Enos  lived   905    years  :   how  many  days  did 
he  live,  allowing  365  days  to  the  year? 

9.  A  planter  raised   208  bales   of  cotton,  each 
bale  weighing  475   pounds  :    how  many,  pounds   of 
cotton  did  he  raise  ? 

10.  If    a    garrison    of    soldiers     consume    4865 
pounds    of  bread    a    day,    how    many    pounds   will 
supply  the  garrison  408  days  ?     606  days  ? 

11.  What  will  508  horses  cost,  at  $125  apiece? 

12.  What  will  it  cost  to  build  705  miles  of  rail- 
road, at  $7525  a  mile? 


60  INTERMEDIATE    ARITHMETIC^ 

LESSON    V. 

MENTAL    EXERCISES. 

JVew  Multiplicand  Figure,  8. 

1.  Three  times  8  are  how  many?     5  times  8? 

7  times  8?     9  times  8?     8  times  8? 

2.  How  many  are  5   times  8  ?     8  times  5  ?     6 
times  8?     8  times  6?     7  times  8?     8  times  7? 

3.  There  are  8  quarts  in  one  peck :    how  many 
quarts  in  3  pecks  ?     5  pecks  ?     7  pecks  ? 

4.  There   are   8   pints  in   a  gallon :    how   many 
pints  in  4  gallons?     6  gallons?     8  gallons? 

5.  If  5   men    can   mow  a   field   of  grass   in   8 
days,  how  long  would  it  take  one  man  to  do  it? 

6.  If  a  quantity  of  provisions  will  last  7  men 

8  days,  how  long  will  it  last  one  man? 

7.  If  4  pipes  will  empty  a  cistern  in  8  hours, 
how  long  will  it  take  one  pipe  to  empty  it? 

8.  If  a  man  earn  8  dollars  a  week,  how  much 
will  he  earn  in  9  weeks  ?     11  weeks  ? 

9.  A    railroad    car    has    8    wheels :    how   many 
wheels  has  a  train  of  7  cars  ?     9  cars  ? 

10.  If  a  horse    eat  8    quarts   of  oats    each   day, 
how  much  will  he  eat  in  6  days  ?     10  days  ? 

11.  If  a   pint  of  oil  cost   8   cents,  what  will   8 
pints  cost? 

12.  James  has  8  marbles,  and  John  has  6  times 
as  many :   how  many  marbles  has  John  ? 

13.  What   will    8    pounds    of   beef   cost,    at   10 
cents  a  pound? 


MULTIPLICATION.  61 

WRITTEN    EXERCISES. 

Multiplicand   or   Multiplier  ending  with  Ciphers. 
1.  Multiply  148000  by  47. 

PROCESS.  To  shorten  the  process,  write  the 

148000  multiplier  under  the  significant  fig- 

47  ures  of  the  multiplicand,  and,  omit- 
ting  the    ciphers    in    forming   the 

Partial     j1  partial    products,    annex    them    to 

Products,  (592  the  product  obtained.     The  result 

Product,     6956000  wm  be  the  true  product. 


NOTE.  —  The  teacher  should  show  that  the  use  of  the 
ciphers  in  forming  the  partial  products  would  produce  the 
same  result. 

(2)  (3)  (4)  (5) 

48000  308000  295  4306 

36  405  43000  245000 


6.  There  are  5280  feet  in  a  mile:    how  many 
feet  in  805  miles? 

7.  The   earth  moves  in  its  orbit  at  an  average 
rate    of    68000    miles   in    an   hour :    how   far   does 
it  move  in  24  hours?     In  48  hours? 

8.  If   a    carriage-wheel    revolve    280    times    in 
running  a  mile,  how  many  times  will  it  revolve  in 
running  68  miles?     75  miles? 

9.  A  canal-boat  was  loaded  with  245  bales  of 
hay,    weighing    280   .pounds    each :    what   was    the 
weight  of  the  cargo  ? 

10.   There  are   480   sheets    of  paper  in  a  ream  : 
how  many  sheets  are  there  in  604  reams? 


62  INTERMEDIATE    ARITHMETIC. 

11.  If-  an  acre  of  land  produce   380  pounds  of 
cotton,  how  many  pounds  will  a  plantation   of  248 
acres  produce  ? 

12.  A  steamboat  makes  145   trips  in  a  season, 
and    carries,  on    an   average,  280    passengers    each 
trip  :   how  many  passengers  does   she  carry  during 
the  season  ? 

LESSON   VI. 

.  MENTAL    EXERCISES. 
New  Multiplicand  Figure,  9. 

1.  Three    times    9    are    how    many  ?     4    times 
9?     6    times    9?     8    times    9?     7    times    9?     9 
times   9? 

2.  How  many  are   5   times   9?     9  times  5?     7 
times  9?     9  times  7?     10  times  9?     9  times  10? 

3.  How  many  are    5    times  10?     10    times    5? 
7    times    10?     10     times     7?     9     times     10?     10 
times  9  ? 

4.  A  man    gave    7  boys    9   raisins   each :    how 
many  raisins  did  he  give  them  all  ? 

5.  If  a  man  earn  10  dollars  a  week,  how  much 
will  he  earn  in  8  weeks  ? 

6.  Jane    writes    9    lines   each    day    at    school : 
how  many  lines  does  she  write  in  8  days  ? 

7.  Charles  receives  9  dollars  a  month  as  errand- 
boy  :   how  much  will  he  earn  in  10  months  ? 

8.  If  7  men  can  do  a  piece  of  work  in  9  days, 
how  many  men  will  it  take   to   do   the  same  work 
in  one  day  ? 


MULTIPLICATION.  63 

9.    If  a   quantity   of  provisions   will   supply  10 
men  9  days,  how  long  will  it  supply  one  man  ? 

10.  What   will    6    barrels    of   flour   cost,    at    $9 
a  barrel  ? 

WRITTEN    EXERCISES. 

Both   Multiplicand  and  Multiplier  ending 
Ciphers. 


1.  Multiply  198000  by  8900. 

PROCESS.  Write  the  significant  figures 

of  the  multiplier  under  the  sig- 
nificant figures'  of  the  multipli- 
cand, and  multiply,  omitting 
Partial  j  1782  the  ciphers  in  forming  the 


MI* 


Products,  I  1 5  8 4  partial   products,  but  annexing 

Product,      1762200000         them   to  the   product  obtained 

for  the  true  product. 

(2)  (3)  (4)  (5) 

94000  90800  470000  950000 

1600  370000  1900  360000 


6.  There   are   8600   seconds   in   one  hour :   how 
many  seconds  are  there  in  630  hours  ? 

7.  Light  moves  192000  miles  in  a  second :  how 
far  does  it  move  in  one  hour  ? 

8.  A   ship  has   provisions  enough  to    allow  the 
crew  130  pounds   a   day    for   90   days :    how   many 
pounds  of  provisions  are  aboard? 

9.  What  will  1700   tons  of   railroad    iron   cost, 
at  $250  a  ton? 

10.    An    army    is    composed    of    54    regiments, 


64  INTERMEDIATE    ARITHMETIC. 

containing,  on    an    average,  670    men    each:    how 
many  men  in  the  army  ? 

11.  If    a    steamer    can    run    260    miles    a    day, 
how  far  can  it  run  in  10  days?     In  100  days? 

12.  In  a  field   of  corn  there    are   70  rows,  and 
each  row  contains  280  hills,  and  each  hill  3  stalks  : 
how  many  stalks  of  corn  in  the  field? 

LESSON    VII. 


1.  What  will  4  oranges  cost,  at  5  cents  apiece? 

2.  What  will    5   barrels    of  flour  cost,   at  $9  a 
barrel  ? 

3.  If  an   orange   is  worth  5   apples,  how  many 
apples  are  7  oranges  worth  ? 

4.  If  there  are  8  pints  in  a  gallon,  how  many 
pints  are  there  in  6  gallons  ? 

5.  Two    men   start   from   the    same    place,    and 
travel  in  opposite  directions,  one  at  the  rate  of  3 
miles    an    hour   and    the    other   4    miles    an    hour  : 
how  far  will  they  be  apart  in  8  hours? 

6.  If  an  orange  is  worth  2  lemons  and  a  lemon 
is    worth    5    plums,    how    many    plums    are    worth 
6    oranges  ? 

7.  If  7  men  can  do  a  piece  of  work  in  5  days, 
how  long  would  it  take  one  man  to  do  it  ? 

8.  If  6  men  can  cut  a  field  of  grass  in  8  days, 
how  many  men  will  it  take  to  cut  it  in  .  one  day  ? 

9.  If  3  pipes  fill  a  cistern  in  10  hours,  in  how 
many  hours  will  one  pipe  fill  it? 


MULTIPLICATION.  65 


WRITTEN    EXERCISES. 

1.  What  is  the  product  of  4894  X  37? 

2.  What  is  the  product  of  5680  X  340? 

3.  6084  X  3008  ==  how  many? 

4.  704000  X  4800  *  =  how  many? 

5.  Multiply   forty-eight   thousand  by  sixty-five 
thousand. 

6.  A    freight    train    consists    of    37    cars,    and 
each    car   contains    9850    pounds    of  freight :    how 
much  freight  in  the  entire  train  ? 

7.  If  980  pounds  of  bread  will   supply  the  in- 
mates   of   the    State    Prison    one    day,    how    many 
pounds  will  supply  them  one  year,  or  365  days  ? 

8.  If  a  sack  of  salt  contain  168  pounds,  what 
will  be  the  weight  of  1600  sacks  ? 

9.  A   merchant    bought    18    firkins    of    butter, 
each  weighing    32    pounds,  at   37   cents   a   pound : 
what  did  it  cost  ? 

10.  A  train  of  27  cars  is  loaded  with  iron ;  each 
car  contains  48  bars,  and  each  bar  weighs  365 
pounds ;  what  is  the  weight  of  the  cargo  ? 


DEFINITIONS,  PEINOIPLES,  AND   EULE. 

Art.  33.  Multiplication  is  the  process  of 
taking  one  number  as  many  times  as  there  are 
units  in  another. 

The  Multiplicand  is  the  number  taken  or 
multiplied. 

I.  A.— 5. 


66  INTERMEDIATE     ARITHMETIC. 

The  Multiplier  is  the  number  denoting  how 
many  times  the  multiplicand  is  taken. 

The  Product  is  the  number  obtained  by  mul- 
tiplying. 

The  multiplicand  and  multiplier  are  called  the 
Factors  of  the  product. 

Art.  34.  The  Sign  of  Multiplication  is  X ,  and 
is  read  multiplied  by.  When  placed  between  two 
numbers,  it  shows  that  the  number  before  it  is 
to  be  multiplied  by  the  number  after  it.  Thus : 
6  X  3  is  read  6  multiplied  by  3. 

NOTE.  —  Since  a  change  in  the  order  of  the  factors  does  not 
change  the  product,  6  X  3  may  also  be  read  6  times  3. 

Art.  35.  Multiplication  is  a  short  method  of 
addition.  The  sum  of5  +  5-f5  +  5  is  the  same 
as  4  times  5. 

Art.  36.  RULE  FOR  MULTIPLICATION.  —  1.  Write 
the  multiplier  under  the  multiplicand,  placing  units 
lender  units,  tens  under  tens,  etc. 

2.  When  the  multiplier  consists  of  but  one  term, 
begin   at   the    right    and    multiply    successively    each 
term    of   the    multiplicand,    writing    the    right-hand 
term  of  each  result  in  the  product  and  adding  the 
left-hand  term  to  the  next  result. 

3.  When  the  multiplier  consists  of  more  than  one 
term,  multiply  the  multiplicand  successively  by  each 
significant  term   of  the  multiplier,  ivriting  the  first 
term    of  each  partial    product   under  the    term    of 
the  multiplier  which  produces  it. 


MULTIPLICATION.  67 

4.  Add  the  partial  products  thus  obtained,  and 
the  sum  will  be  the  true  product. 

Art.  37.  1.  When  the  multiplier  or  multiplicand 
or  both  end  with  one  or  more  ciphers,  omit  the 
ciphers  in  the  partial  products  and  annex  them 
to  the  product  obtained. 

2.  Any  number  may  be  multiplied  by  10,  100, 
1000,  etc,  by  annexing  to  it  as  many  ciphers  as 
there  are  ciphers  in  the  multiplier. 

LESSON   VII. 

MENTAL    EXERCISES. 

Problems  combining  Addition  9  Subtraction,  and 
Multiplication. 

1.  6x7  +  4  +  5+8  +  7—6—  how  many ? 

2.  8x4+6  —  3  +  2  —  5  +  6—  how  many? 

3.  A   grocer  bought   10    barrels    of   apples,   at 
$4   a   barrel,    and    sold   them    so    as   to   gain    $15 : 
for  how  much  did  he  sell  them  ? 

4.  John  has  6  marbles,  and  Willis  has  4  times 
as  many  less  9,  and  Charles  has  as  many  as  both 
John  and  Willis :    how  many  marbles  has  Charles  ? 

5.  A   lady   teacher   receives    $9    a    week,    and 
spends  $6  for  board   and  washing :   how  much   can 
she  save  in  8  weeks  ? 

6.  Two    men    start   from    the    same    place    and 
travel   in    opposite    directions,   one    at    7   miles    an 
hour  and  the    other  at   5   miles  an  hour :    how  far 
will  they  be  apart  in  8  hours? 


68  INTERMEDIATE     ARITHMETIC. 

7.  Two    stages   start  from  the    same    place  and 
go  in  the  same  direction,  one  at  9   miles  an  hour 
and   the    other  at    6    miles  an   hour :    how  far  will 
they  be  apart  in  5  hours  ? 

8.  When    oranges    are    sold   at    7    cents    apiece 
and   lemons    at    5    cents    apiece,   how   many    cents 
will  buy  6  oranges  and  8  lemons  ? 

9.  If  a  man  earn  $8  a   week   and   a   boy   $3, 
how  much  will  they  both  earn  in  7  weeks  ? 

10.  A  pedestrian  left  a  city  and  walked  9  hours 
at  the    rate  of  4  miles  an  hour ;   he  then  returned 
at   the   rate   of  3    miles   an    hour,    but  in   4   hours 
stopped  to  rest :   how  far  was  he  from  the  city  ? 

11.  If  a   man   earn  $12   a   week  and   spend   $7, 
how  much  will  he  save  in  9  weeks  ? 


WRITTEN    EXERCISES. 

1.  From  4080  X  26  take  2024  X  16. 

2.  A   grocer    bought    275    barrels    of   flour  for 
$2475,   and  sold  it  at  $12   a  barrel :   what  did  he 
gain  ? 

3.  A    clerk    receives    $125    a    month,    and    his 
expenses  are  $68  a  month :   how  much  does  he  lay 
up  each  year? 

4.  An  agent  sold   48  sets  of  outline  maps,  at 
$16   a  set ;    the    maps  cost  him    $10    a    set :    how 
touch  did  he  make? 

5.  If    a    steamer    carry,    on     an    average,    75 
passengers  each  trip,  how  many  passengers  will  it 
carry  in  12  weeks,  making  3  trips  a  week  ? 


MULTIPLICATION.  69 

6.  A    book     contains    288     pages,    each    page 
contains   42    lines,   and    each    line  13    words :    how 
many  words  in  the  book  ? 

7.  A  man  bought  a  farm  for  $4780 ;   he   sold 
80  acres   at  $33  an   acre,  and  the  remaining    por- 
tion for  $2560.     How  much   did   he   make   by  the 
transaction  ? 

8.  A    regiment    contains    960    men,    exclusive 
of   the    commissioned    officers ;     the    men    receive 
$16    a    month,    and   the    aggregate    salary    of    the 
officers   is   $2800   a  month.     What  is   the   monthly 
pay   of  the   regiment  ? 

9.  A    drover    bought    480    head    of    cattle    in 
Ohio,  at  $45   a  head,  shipped  them   to  New  York, 
at  an  expense  of  $6   a  head,  and  then   sold  them 
at  $56  a  head.     How  much  did  he  make  ? 

10.  A  miller  manufactured  560  barrels  of  flour, 
and  sold  it  at   $9  a  barrel ;   the  wheat  cost  $2750, 
and   the   expense    of   running  the   mill  was   $960 : 
how  much  did  he  make  ? 

11.  A  man    sold   5  horses   at    $87   apiece,   and 
received    $350    in    cash    and   a    note    for   the   bal- 
ance :   what  was  the  value  of  the  note  ? 

12.  The   President's    salary    is    $25000    a   year: 
if  his    expenses    are    $1500    a    month,    how    much 
can  he  save  during  his  term  of  4  years  ? 

13.  If  a   quantity  of  provisions  will  supply  960 
soldiers   27    days,   how  many    soldiers   will   it   sup- 
ply one'  day? 


70  INTERMEDIATE    ARITHMETIC. 


SECTION     V. 


LESSON   I. 
3)irisor  Figures,   1 ,  2,  3. 

1.  How  many  times  is  2  contained  in  6?     2  in 
12?     2  in  16?     2  in  18?     2  in  20? 

2.  How  many  times  is  3  contained  in  9  ?     3  in 
12?     3  in  15?     3  in  18?     3  in  21  ?     3  in  27? 

3.  How  many  times  is   2    contained   in   12?     3 
in  12?     2  in  18?     3  in  18?     2  in  24  ?     3  in  24? 

4.  Two  boys  sit  at  one  desk :    how  many  desks 
will  seat  8  boys?     16  boys? 

5.  If  a   man   walk   3   miles   an   hour,   how  long 
will  it  take  him  to  walk  15  miles?     21  miles? 

6.  William  having  20  plums,  divided  them  among 
his    companions,    giving    2    to    each :     how    many 
companions  had  he  ? 

WRITTEN    EXERCISES. 

1.   Divide   848  by  2. 

PROCESS.  Write  the   divisor   at   the 

left  of  the  dividend,  and  draw 
Divisor,   2)848,   Dividend,  ,  ,.       .  . 

a  curved  line  between  them, 
424,   Quotient,          and  a  straight  line  under  the 

dividend.     Begin  at  the  left, 

and  divide  successively  each  term  of  the  dividend  by  the 
divisor.     The  quotient  is  424. 


DIVISION.  71 

(2)  (3)  (4)  (5)  (6) 

2)482          2)8642          3)6936          3)9369          3)3696 

7.  Divide  3609  by  3.     8084  by  2. 

8.  Divide  4684  by  2.     6309  by  3. 

9.  At  $3  a  bushel,  how  many  bushels  of  wheat 
can  be  bought  for  $963?     For  $639? 

10.  In   how   many   hours    can   a   man    walk    396 
miles,  if  he  walk  at  the  rate  of  3  miles  an  hour? 

11.  If  a   man   earn   $2   a  day,  how  long  will  it 
take  him  to  earn  $360  ? 

LESSON    II. 

MENTAL    EXERCISES. 
New  ^Divisor  Figures,  &.  and  6. 

1.  How  many  times   is   4   contained  in   8  ?     4 
in  12  ?     4  in  20  ?     4  in  28  ?     4  in  36  ? 

2.  How  many  5's  in  15?    30?    40?    25?    50? 
35?    45?    20? 

3.  In  an    orchard  there    are  16  trees,  standing 
in  rows  of  4  trees  each :   how  many  rows  of  trees 
in  the  orchard? 

4.  How   many   ranks    of   4   soldiers    each  will 
24  soldiers  make  ?     32  soldiers  ?     40  soldiers  ? 

5.  A    man    planted    30    peach    trees    in    rows, 
setting   5   trees  in   each   row  :   how  many  rows  did 
they  make? 

6.  A  school-room   contains   35   desks,  arranged 
with    5    desks   in    each    row :    how   many   rows    of 
desks  in  the  room? 


72  INTERMEDIATE    ARITHMETIC. 

7.  How    many    chairs,    at    $4    apiece,    can    be 
bought  for  $36  ? 

8.  How  many   pairs    of   boots,    at    $5    a    pair, 
can  be  bought  for  $35  ? 

9.  Mary   is    reading    5    chapters    a    day:    how 
long  will  it  take  her  to  read  45  chapters  ? 

10.  A  boy  had  50  peach-stones,  which  he 
planted  in  rows  of  5  each :  how  many  rows  did 
he  plant? 

WRITTEN    EXERCISES. 

1.   Divide  784  by  4. 

PROCESS.  Write  the   divisor  at  the 

j\~,        n-  -i    j         left  of  the  dividend.     Begin 
Divisor,   4 )  784,   Dividend,  ,   ,,      .  ^  .       .  A  Tr. 

at  the  left-hand  term  of  the 

196,   Quotient.          dividend,  and   divide,   thus: 
4  is  contained  in  7  hundreds 

1  hundred  times,  with  3  hundreds  remaining.  Write  the 
1  hundred  in  hundreds'  place  in  the  quotient,  and  re- 
duce the  3  hundreds  remaining  to  30  tens,  which,  with 
the  8  tens  added,  make  38  tens.  Four  is  contained  in 
38  tens  9  ten  times,  with  2  tens  remaining.  Write  the 
9  tens  in  tens'  place  in  the  quotient,  and  reduce  the  2 
tens  remaining  to  20  units,  which,  with  the  4  units 
added,  make  24  units.  Four  is  contained  in  24  units 
6  times.  Write  the  6  units  in  units'  place  in  the  quo- 
tient. The  quotient  is  196. 

(2)  (3)  (4)  (5)  (6) 

4)764  4)936  5)640  5)870  5)765 

7.  Divide  1128  by  2 ;   by  3 ;   by  4. 

8.  Divide  8740  by  2 ;   by  4;   by  5. 


DIVISION.  73 

9.   Divide  18480  by  2 ;   by  3;    by  4 ;   by  5.4 

10.  A  manufacturer  packed  372  clocks  in  boxes, 
placing    4    clocks   in    each   box :    how   many  boxes 
were  required  ? 

11.  If  4  bushels  of  wheat  will  make  a  barrel  of 
flour,  how  many  barrels  will  972  bushels  make  ? 

12.  If  a   man   earn   $4   a   day,   how  many  days 
will  it  take  him  to  earn   $1584? 

LESSON   III. 

MENTAL    EXERCISES. 
JVew  ^Divisor  Figures,   6  and  7 '• 

1.  Six   is    contained    in   12   how   many   times? 

6  in  24?     6  in  36?     6  in  48?     6  in  54? 

2.  How   many  times   7   in  7  ?     7  in  21  ?     7  in 

35  ?     7  in  49  ?     7  in  63  ?     7  in  42  ?     7  in  56  ? 

3.  How   many   6's   in   42?     7's   in  42?     6's  in 
30?     5's  in  30?     7's  in   28?     4's   in   28?     7's  in 
35?     5's  in  35?     7's  in  56? 

4.  If  6  chairs  make  a  set,  how  many  sets  will 

36  chairs  make?     48  chairs?     60  chairs? 

5.  There    are    7   days    in    a   week :    how   many 
weeks  in  49  days?     In  56  days?     63  days? 

6.  There   are   6  feet   in    a    fathom :    how  many 
fathoms  in  54  feet?     In  60  feet? 

7.  An   orchard   contains   56   trees   in    rows   of 

7  trees    each :    how    many    rows    of   trees    in    the 
orchard  ? 

8.  How    many     plows,    at     $6     each,    can    be 
bought  for  $48?     For  $54? 


74  INTERMEDIATE    ARITHMETIC. 

^9.  If  a  horse  travel  6  miles  an  hour,  how 
long  will  it  take  him  to  travel  48  miles  ? 

WRITTEN    EXERCISES. 

1.  Divide  1608  by  67. 

PROCESS.  Draw    a    curved    line    at    the 

67 )  1608  ( 24,  Quotient.        Hght  °f  the  dividend>  to  seParate 
.jg4  it  from   the  quotient.      Since  67 

is  not  contained   in  the   number 
denoted  by  the  first  two  left-hand 

9PQ 

figures  of  the  dividend,  take  160 
tens  for  the  first  partial  dividend. 

Divide  160  tens  by  67,  and  write  the  result  (2  tens)  at 
the  right,  for  the  tens'  figure  of  the  quotient.  Multiply 
67  by  this  quotient  figure,  and  subtract  the  product  (134 
tens)  from  160  tens.  The  remainder  is  26  tens,  to  which 
annex  the  8  units  for  a  second  partial  dividend.  Divide 
268  by  67,  and  write  the  result  (4  units)  for  the  units' 
figure  of  the  quotient.  Multiply  67  by  this  quotient  fig- 
ure, and  subtract  the  product  (268  units)  from  268  units. 
The  quotient  is  24. 

NOTE.  —  It  is  sometimes  difficult  to  tell  how  many  times 
the  divisor  is  contained  in  a  partial  dividend.  When  this  is 
(he  case,  take  the  number  denoted  by  the  first  left-hand  figure, 
or  first  two  left-hand  figures,  of  the  divisor  for  a  trial  divisor, 
and  the  number  denoted  by  the  proper  number  of  left-hand 
figures  in  the  partial  dividend  as  a  trial  dividend.  If  the 
divisor,  multiplied  by  the  value  of  the  quotient  figure  thus 
found,  gives  a  product  greater  than  the  partial  dividend,  the 
quotient  figure  is^  too  large,  and  should  be  reduced.  The  trial 
divisor  in  the  above  example  is  6,  the  first  trial  dividend  is  16, 
and  the  second  26.  The  teacher  should  make  this  process  plain 
to  the  pupil. 

2.  Divide  312  by  24.     374  by  17. 

3.  Divide  792  by  36.     1625  by  65. 


DIVISION.  75 

4.  Divide  2520  by  36.     3024  by  63. 

5.  Divide  64347  by  267.     49179  by  507. 

6.  There  are   36  inches  in  a  yard  :    how  many 
yards  are  there  in  792  inches  ? 

7.  A  bushel    of  corn   weighs    56   pounds :   how 
many  bushels  of  corn  in  24416  pounds  ? 

8.  A   hogshead    of   molasses    contains    63    gal- 
lons :    how  many  hogsheads  in  4788  gallons  ? 

9.  If  72   books   can  be  packed  in  a  box,  how 
many  boxes  will  it  take  to  hold  17496  books? 

10.  How  many  farms  of  156  acres  each,  can  be 
sold  from  a  tract  of  land  containing  7332  acres  ? 

11.  If  a  vessel  sail,  on    an  average,  47  miles  a 
day,  how  long  will  it  take  it  to  sail  2303  miles  ? 

12.  There    are    365    days    in    a    common    year : 
how  many  years  are  there  in  90155  days  ? 

LESSON    IV. 

MENTAL    EXERCISES. 
JVew  ^Divisor  Figure,   8. 

1.  How  many   times   is   8    contained   in   8?     8 
in  24  ?     8  in  40  ?     8  in  56  ?     8  in  72  ? 

2.  How  many  8's  in   56?     7's   in   56?     8's  in 
48?     6's  in  48?     8's  in  72?     9's  in  72? 

3.  32  -r-  8  =  how  many  ?     49  H-  7  ?     54  -f-  6  ? 
64-^8?     56-^-7?     56  -+-  8? 

4.  There   are   8   quarts   in   a   peck :    how  many 
pecks  in  72  quarts  ? 

5.  If  a  steamer  run  8  miles  an  hour,  in  how 
many  hours  will  it  run  80  miles? 


76  INTERMEDIATE    ARITHMETIC. 

6.  There  are   8  furlongs  in  a  mile  :    how  many 
miles  in  56  furlongs? 

7.  At   $8   a  barrel,  how  many  barrels  of  flour 
can  be  bought  for  $64? 

8.  If    a    man    work    8    hours    a    day,    in    how 
many  days  will  he  work  96  hours  ? 

WRITTEN    EXERCISES. 
The  Quotient  containing  One  or  more  Ciphers. 

1.  Divide  34137  by  84. 

PROCESS.  Since  the  divisor  is  not  con- 

84)34137(406  tained  in  the  second  partial  divi- 

c>og  dend   (53    tens),  write   0   in    the 

— — —  tens'  place  in  the  quotient,  and 

annex   the    7    units   for   a   third 
partial  dividend.     As  there  is  no 
33,   Remainder.         figure    of    tne    dividend    left    to 
annex  to  33  to  form  a  new  par- 
tial  dividend,    33    remains    undivided,    arid    is    called   the 
remainder. 

2.  Divide  24399  by  48.     467034  by  806. 

3.  Divide  2845007  by  5728.     215607  by  18036. 

4.  Divide  1423685  by  6785.     1604083  by  2088. 

5.  In    one    week    there    are    168    hours:    how 
many  weeks  in  36248  hours  ? 

6.  A   drover   went   West  with   $23450   to   buy 
cattle :    how  many   cattle    could  he  buy,   at  $58  a 
head  ? 

7.  If  a  garrison  consume  648  pounds  of  bread 
in  a  day,  how  long  will  19608  pounds  last  it? 


DIVISION.  77 

8.  If   the    average    daily   receipts    of   a   ferry- 
boat are  $275,  in  how  many  days  will  the  receipts 
amount  to  $10000? 

9.  How   long  will   it  take   a   pipe,   discharging 
158    gallons    of    water    in    an    hour,    to    empty    a 
cistern  containing  7584  gallons? 

10.  A  cord  of  wood  contains  128  solid  feet : 
how  many  cords  in  a  pile  containing  5280  solid 
feet? 

LESSON  V. 

MENTAL     EXEKCISES. 
New  ^Divisor  Figure,  9. 

1.  How   many   times    9   in    18  ?     9   in   27  ?     9 
in  36?     9  in  54?     9  in  72?     9  in  90  ? 

2.  How  many  9's  in  45  ?     5's  in  45  ?     9's  in 
63?     7's  in  63?     9's  in  72?     8's  in  72? 

3.  How  long  will  it  take  a  steamer  to  make  a 
trip  of  81  miles,  if  it  run  9  miles  an  hour? 

4.  If  9  words  fill   a  line,   how  many  lines  will 
72  words  fill?     81  words? 

5.  If   a   man   can   do    a  piece   of  work   in   90 
days,  how  many  men  can  do  it  in  9  days? 

6.  If  a  quantity  of  provisions  will  last  72  men 
one  day,  how  long  will  it  last  9  men? 

7.  How  many   sheep,    at    $9    a    head,    can    be 
bought  for  $54  ?     For  $63  ? 

8.  A    copy-book    contains    100    lines    with    10 
lines    on    each    page :    how    many    pages    in    the 
book? 


78 


INTERMEDIATE    ARITHMETIC. 


9.   If  a  man   earn   $10   a  week,  how  long  will 
it  take  him  to  earn  $100  ? 

10.    How    many    tons    of    hay,    at    $10    a    ton, 
can  be  bought  for  $90. 


WKITTEN    EXERCISES. 
The  ^Divisor  ending  in  One  or  more  Ciphers. 


1.  Divide  350  by  10. 

FIRST    PROCESS. 

10)350(35 
30 


SECOND    PROCESS. 

1|0)35|0 

35,   Quotient. 

2.   Divide  2865   by  100. 

1|00)28!65 

28,         Quotient. 
65,   ^Remainder. 


By  comparing  these 
two  processes,  it  is  seen 
that  350  is  divided  by 
10,  by  cutting  off  the 
right-hand  figure.  The 
reason  is  obvious.  The 
cutting  off  of  the  right- 
hand  figure  removes  each 
of  the  other  figures  one 
place  to  the  right,  and 
thus  decreases  their  value 
ten-fold.  In  like  man- 
ner, it  may  be  shown 
that  cutting  off  the  two 
right-hand  figures  divides 
a  number  by  100 ;  cut- 
ting off  three  right-hand 
figures,  by  1000,  etc. 


3.  Divide  45600  by  10.     By  100. 

4.  Divide  187000  by  1000.     By  100. 

5.  Divide  384050  by  100.     By  1000. 

6.  Divide  230045  by  1000.     By  10000. 

7.  Divide  450860  by  10000.     By  1000. 


DIVISION.  79 

8.   Divide  196800  by  4800. 

PROCESS.  First  divide  both  divisor 

and  dividend  by  100,  which 

48100)1968100(41,   Quotient.       is  done  by  cutting   off  the 

*"*  two  right-hand  figures.  Then 

48  divide  1968,   the   new    divi- 

48  dend,  by  48,  the  new  divisor. 

The  quotient  is  41. 


E.  —  The  teacher  can  easily  show  that  both  divisor  and 
dividend  may  be  divided  by  100  (or  any  other  number)  without 
affecting  the  value  of  the  quotient. 

9.  Divide  63200  by  7900. 

10.  Divide  116087000  by  2900. 

11.  Divide  70338000  by  75000. 

12.  Divide  58864  by  4500. 

PROCESS.  First  divide  both  divisor  and 

dividend  by  100,  which,  in  the 
45!00)588|64(13  case  of  the  dividend>  leaves   a    * 

4°  remainder  of  64.     Next   divide 

138  548  by  45,  leaving  a  remainder 

135  of  3   hundreds,  to  which   annex 

364,   Remainder,      the  first  remainder  (64),  obtain- 

ing 364  for  the  true  remainder. 

NOTE.  —  The  true  remainder  is  found  by  annexing  the  first 
remainder  to  the  second.  The  reason  for  this  can  be  easily 
given  by  the  teacher. 

13.  Divide  466384  by  3900.     220345  by  940. 

14.  Divide  99990  by  5400.     172800   by  14400. 

15.  A  barrel  of  beef  contains  200  pounds  :  how 
many  barrels  will  contain  12800  pounds? 

16.  There   are  480   sheets   of  paper  in  a   ream  : 
how  many  reams  will  129600  sheets  make? 


80  INTERMEDIATE    ARITHMETIC. 

17.  There    are    3600    seconds   in    an    hour :    how 
many  hours  in  172800  seconds? 

18.  How    many   city    lots,    at    $1600    each,    can 
be  bought  for  $25600? 

19.  How  many  cars,  each  carrying  1800  pounds, 
will   transport   723690   pounds   of  hay? 

20.  How  many  barrels,  each  holding  196  pounds, 
will  hold   8450  pounds  of  flour? 

21.  How  many    regiments,    averaging    750    men 
each,  will  make  an  army  of  35000  men  ? 

22.  A   peach   orchard   contains   6758   trees,   and 
there   are,   on   an   average,  62  trees  on  each  acre : 
how  many  acres  in  the  orchard  ? 

23.  A   pipe   discharges   94   gallons    in    an   hour : 
in  how  many  hours  will  it  empty  a  cistern  holding 
3384  gallons  of  water  ? 

24.  What  number  multiplied  by  98  will  produce 
15288? 

'  25.    The  dividend  is  5292  and  the  divisor  is  63 : 
what  is  the  quotient  ? 

26.  The  divisor  is  $1500  and  the  dividend 
$564000:  what  is  the  quotient? 

DEFINITIONS,  PKINOIPLES,  AND  KULES, 

Art.  38.  Division  is  the  •  process  of  finding 
how  many  times  one  number  is  contained  in 
another. 

The  Dividend   is   the   number  divided. 

The  Divisor  is  the  number  by  which  the 
dividend  is  divided. 


DIVISION.  81 

The  Quotient  is  the  number  of  times  the 
divisor  is  contained  in  the  dividend. 

The  Remainder  is  the  part  of  the  dividend 
which  is  left  undivided.  When  the  dividend  con- 
tains the  divisor  an  exact  number  of  times,  there 
is  no  remainder. 

Art.  39.  The  Sign  of  Division  is  -T-,  and  is 
read  divided  by.  When  placed  between  two  num- 
bers, it  shows  that  the  number  before  it  is  to  be 
divided  by  the  number  after  it.  Thus :  16  -4- 
4  ==  4  is  read  16  divided  by  4  equals  4. 

Division  is  also  expressed  by  writing  the  dividend  above 
and  the  divisor  below  a  short  horizontal  line.  Thus :  *-/  is 
read  18  divided  by  3. 

Art.  40.  One  number  is  contained  in  another 
number  as  many  times  as  it  can  be  taken  from  it. 
Hence  division  may  be  regarded  as  a  short  method 
of  subtraction. 

A  number  is  contained  in  another  as  many 
times  as  it  must  be  taken  to  produce  it.  Hence 
division  may  be  regarded  as  the  reverse  of  mul- 
tiplication. The  divisor  and  quotient  are  factors 
of  the  dividend. 

Art.  41.  There  are  two  methods  of  division,  called 
Short  Division  and  Long  Division. 

In  Short  Division,  the  partial  products  and 
partial  dividends  are  not  written,  but  are  formed 
mentally.  This  method  is  generally  used  when 
the  divisor  does  not  exceed  12. 

In  Long  Division,  the  partial  products  and 
partial  dividends  are  written. 

I.  A.— 0. 


82  INTERMEDIATE     ARITHMETIC. 

Art,  42.  RULE  FOR  SHORT  DIVISION.  —  1.  Write 
the  divisor  at  the  left  of  the  dividend,  and  draw 
a  curved  line  between  them  and  a  straight  line 
under  the  dividend. 

2.  Find  hoiv  many  times  the  divisor  is  contained 
in    the    left-hand    term    or    terms    of   the    dividend, 
taken    as   a  partial    dividend,    and    write    the    quo- 
tient under  the  last  figure   of  the   dividend  used. 

3.  Multiply    the    divisor    by    the    quotient    term 
found,   and   subtract  the  product  from   the  partial 
dividend  used,  performing  each  process  mentally. 

4.  Prefix    the   remainder,   if  there    be  one,   to  the 
next    term    of   the    dividend   for    a    second   partial 
dividend,    and    divide,    multiply,    and    subtract    as 
before. 

5.  Proceed    in   this    manner   until    all   the   terms 
of  the  dividend  have  been  used. 

PROOF.  —  Multiply  the  divisor  by  the  quotient, 
to  the  product  add  the  remainder,  if  there  be  any, 
and  if  the  result  equals  the  dividend,  the  work 
is  correct. 

Art.  43.  RULE  FOR  LONG  DIVISION.  —  1.  Write 
the  divisor  at  the  left  of  the  dividend,  and  draw 
a  curved  line  between  them,  and  also  at  the  right 
of  the  dividend,  to  separate  it  from  the  quotient. 

2.  Take  as  many  of  the  left-hand  terms  of  the 
dividend  as  will  contain  the  divisor,  for  a  par- 
tial dividend;  find  how  many  times  this  will  con- 
tain the  divisor,  and  write  the  quotient  at  the 


DIVISION.  83 

right   of  the    dividend  for    the  first   left-hand    term 
of  the  quotient. 

3.  Multiply    the    divisor    by    the    quotient    term 
found,   write   the  product   under   the   partial    divi- 
dend used)  and  subtract. 

4.  To    the    remainder    annex    the    next    term    of 
the   dividend  for  a    second  partial    dividend,    and 
divide,   multiply,  and  subtract  as  before. 

5.  Proceed    in   this   manner   until    all    the   terms 
of  the  dividend  have   been   used. 

NOTE.  —  When  any  partial  dividend  does  not  contain  the 
divisor,  write  a  cipher  in  the  quotient,  and  annex  another 
term  of  the  dividend  to  form  a  new  partial  dividend. 

Art.  44.  When  one  or  more  of  the  right-hand 
figures  of  the  divisor  are  ciphers  — 

1.  Cut    off   the    ciphers    from    the    right    of   the 
divisor,  and  an  equal  number  of  figures  from  the 
right  of  the   dividend. 

2.  Divide  the  new   dividend  thus  formed  by  the 
new   divisor,  and  the  result  will   be   the  quotient. 

3.  Prefix   the   remainder,  if  there   be   one,  to  the 
figures    cut   off  from    the    dividend,   and   the   result 
will  be  the  true  remainder. 

Art.  45.  To  divide  any  number  by  10,  100, 
1000,  etc.,- 

Cut  off  as  many  figures  from  the  right  as  there 
are  ciphers  in  the  divisor.  The  figures  cut  off  are 
the  true  remainder. 


84  INTERMEDIATE    ARITHMETIC. 

LESSON    VI. 
Miscellaneous   fteriew    'Problems. 

1.  The  sum  of  two  numbers  is  15    and  one  of 
the  numbers  is  6  :    what  is  the  other  ? 

2.  The    difference    between    two    numbers   is    8 
and  the  smaller  number  is  9  :   what  is  the  larger  ? 

3.  The  product  of  two  numbers  is  56  and  one 
of  the  numbers  is  7:    what  is  the   other? 

4.  The    quotient    of    two     numbers    is    6    and 
the  divisor  is  8 :    what  is  the  dividend  ? 

5.  How  many  barrels  of  flour,  at  $8  a  barrel, 
will  pay  for  24  yards  of  carpeting,  at  $2  a  yard  ? 

6.  How  many  tons   of  coal,  at    $9    a   ton,  will 
pay  for  15  cords  of  wood,  at  $6  a  cord? 

7.  A  grocer  bought  7  barrels  of  flour,  at  $6  a 
barrel :    for  how  much   a  barrel   must  he   sell  it  to 
gain  $14  on  the  lot? 

8.  If  one   man   can   build    a   wall    in   36    days, 
how  many. men  can  build  it  in  4  days? 

9.  If   6    men    can   do    a    piece    of   work   in   8 
days,  how  many  men  can  do  it  in  12  days  ? 

10.  Two  steamers  start  from  the  same  port  and 
sail  in  the  same  direction,  one  sailing  12  miles 
an  hour  and  the  other  9  miles  an  hour:  how 
far  apart  will  they  be  in  10  hours  ? 

WRITTEN    EXERCISES. 

1.    The    greater   of  two   numbers   is   4056    and 
their  difference  is  3650  :  what  is  the  less  number  ? 


DIVISION.  85 

2.  The  subtrahend  is  34203  and  the  remainder 
is  8706  :   what  is  the  minuend  ? 

3.  The    divisor   is    534    and    the   quotient    43 : 
what  is  the  dividend? 

4.  The    product  of  two   numbers   is    5328    and 
one  of  the  numbers  is  148 :    what  is  the  other  ? 

5.  Multiply  the   sum   of  486  and  392  by  their 
difference. 

6.  Divide   the   product   of  48   and  24  by  their 
difference. 

7.  A   merchant   bought   35    yards    of   cloth   at 
$56,   and    sold    it    at    $2    a  yard:    how  much   did 
he  gain? 

8.  A  drover  bought  240   sheep   at  $8   a  head, 
and  then    sold   90    of   them  at  $12   a  head,  75   at 
$9    a    head,    and    the    rest    at    $6    a    head.      How 
much   did  he   gain  ? 

9.  A  farmer    exchanges    65    bushels    of   wheat 
at   $2    a   bushel    and   35    sheep    at   $6    a   head  for 
cows    at    $34    a   head.      How    many    cows    did    he 
receive  ? 

10.  A  man's   income  is  $3500  a  year :   he  pays 
$450   a  year  for  house-rent,  $150   for  taxes,  $350 
for   hired    help,    and   $45    a    month    for   other   ex- 
penses.     How   much   has    he   left   at   the    close    of 
the  year  ? 

11.  A  man  bought  80   acres  of  land  at  $35  an 
acre,  paid   $325    for  improvements,  and   then    sold 
it  for  $3750.     How  much  did  he  gain? 

12.  A   grain  merchant  having    3500   bushels    of 
wheat,   sold   1650   bushels,   and  then   bought  twice 


86  INTERMEDIATE    ARITHMETIC. 

as  much    as  he    had  left.      How  many  bushels  did 
he  buy? 

13.  A  man  left  an   estate  to  his  wife   and  three 
children.     The  wife   received  $4500 ;   the  youngest 
child,   $1500 ;    the    second   child,   $1850 ;    and   the 
eldest    child   as   much    as   both    of   the    others    less 
$1350.     What  was  the  value  of  the  estate  ? 

14.  A   and   B    start   together   on   a   journey,  A 
traveling    28    miles   a   day   and  B   33   miles :    how 
far  apart  will  they  be  in  12  days  ? 

15.  A  and  B    start  together   and  travel  in   op- 
posite  directions,  A  going   28   miles   a   day  and  B 
33  miles :   how  far  apart  will  they  be  in  12  days  ? 


QUESTIONS  FOR  KEVIEW. 

What  is  addition?  What  is  meant  by  sum  or  amount? 
What  does  it  contain?  What  is  meant  by  like  numbers? 
What  numbers  can  be  added  ?  What  is  the  sign  of  ad- 
dition? What  does  it  show?  Give  the  rule  for  addition. 
What  is  the  method  of  proof? 

What  is  subtraction?  The  difference  or  remainder? 
The  minuend?  The  subtrahend?  What  numbers  can  be 
subtracted?  What  does  the  sum  of  the  remainder  and 
subtrahend  equal  ?  What  is  the  sign  of  subtraction  ? 
What  does  it  show  ?  Give  the  rule  for  subtraction.  What 
is  a  method  of  proof? 

What  is  multiplication  ?  The  multiplicand  ?  The  mul- 
tiplier? The  product?  Of  what  are  the  multiplicand  and 
multiplier  factors?  What  is  the  sign  of  multiplication? 
What  does  it  show?  How  may  the  product  be  obtained 
by  addition? 

Give  the  rule  for  multiplication.     How  may  you  mul- 


PROPERTIES    OF    NUMBERS.  87 

tiply  when  either  the  multiplicand  or  multiplier,  or  both, 
end  in  ciphers?  How  may  any  number  be  multiplied  by 
10,  100,  1000,  etc.? 

What  is  division?  The  dividend?  The  divisor?  The 
quotient?  The  remainder?  What  is  the  sign  of  division? 
What  does  it  show?  In  what  other  way  may  division  be 
expressed  ?  How  many  times  may  the  divisor  be  sub- 
tracted from  the  dividend  ?  Of  what  is  division  the  re- 
verse ? 

What  is  short  division  ?  When  is  it  used  ?  Give  the 
rule.  What  is  long  division  ?  Give  the  rule.  How  do 
you  proceed  when  a  partial  dividend  will  not  contain  the 
divisor?  How  may  you  divide  when  the  divisor  ends  in 
ciphers?  How  may  any  number  be  divided  by  10,  100, 
1000,  etc.? 


SECTIOIV     VI. 

Of7 


LESSON   I. 
3)ivisor,  Greatest  Common  ^Divisor,  and  Factor. 

NOTE. — The   term   number,   used    in    this    section,    denotes 
an   integer. 

1.  What  numbers   besides  itself  and  1  will  ex- 
actly divide  15?    21?    25?    30?    56?    63? 

2.  What  numbers  besides  itself  and  1  will  ex- 
actly divide  7  ?    11?    13?    17?    23?    37?    41? 

3.  What    numbers    will   exactly    divide    4  ?     5  ? 
16?    19?    24?    29?    33?    31?    42? 


88  INTERMEDIATE    ARITHMETIC. 

4.  What    are    the    divisors    of    10?     28?     31? 
33?    43?    49?    53?    55?    70?    90?    99? 

NOTE.  — Since  every  number  is  exactly  divisible  by  itself 
and  1,  these  divisors  need  not  be  given. 

5.  What  number  is  a  divisor  of  both  9  and  12? 
15  and  20?     24  and  27?     42  and  56? 

6.  What  divisor  is  common  to  28  and  35  ?     27 
and  36?     42  and  54?     63  and  81? 

7.  What    is   a   common    divisor  of  15  and  30? 
45  and  60?     50  and  75?     60  and  84? 

8.  What  is  the   greatest  divisor  common  to  48 
and  72?     27  and  54?     50  and  75? 

9.  What  is  the  greatest  common  divisor  of  24 
and  36?     32  and  48?     56  and  84? 

10.  What   is   a   common   divisor   of  16,   32,    and 
48?     15,  30,  and  45?     36,  54,  and  72? 

11.  What  is  the  greatest  common  divisor  of  32 
and-  48?      15,    30,    and    45?      36,    54,    and     72? 
18,   45,   and   81? 

Art.  46.  A  number,  that  has  no  divisor  except 
itself  and  1,  is  called  a  Prime  Number.  A 
number  that  has  other  divisors  besides  itself  and 
1,  is  called  a  Composite  Number. 

12.  Name  all  the  prime  numbers  between  0  and 
20.     Between  20  and  30.     40  and  50. 

13.  Name    all   the    composite    numbers    between 
20  and  30.     40  and  50.     60  and  70. 

14.  What   are    the    prime    divisors   of  15?     18? 
22?    28?    33?    36?    37?    40?    43? 


PROPERTIES    OF    NUMBERS.  89 

15.  What   are    the    prime    divisors    of  16?     20? 
25?    27?    35?    44?    55?    60? 

Art.  47.  The  divisors  of  a  number  are  called 
its  Factoids;  and  prime  divisors  are  called 
Prime  Factors. 

16.  What   are    the    prime    factors    of   21?     24? 
35?    39?    42?    49?    54?    56?    63?    66?    72? 

17.  Of    what    number    are    2,    3,    and    5    prime 

factors?     2,  5,  and  7?     2,  2,  3,  and  5? 

NOTE.  —  The  product  of  the  prime  factors  of  a  number 
equals  the  number. 

18.  Of  what  number   are   2,  2,  3,  and   3   prime 
factors?     2,  3,  5,  and   7?     3,  5,  2,  and   7? 


•WRITTEN    EXERCISES. 

1.    What  are  the  prime  factors  of  126? 

PROCESS.  Divide  126   by   2,  a  prime 

2)126  divisor;    next  divide  the  quo- 

ow'o  tient  63  by  3,  a  prime  divisor; 

and  then  divide  the  quotient 

21    by    3,    a    prime    divisor. 

7  The   prime    factors    are    2,  3, 

126  =  2  X  3  X  3  X  7.         3,  and  7. 

What  are  the  prime  factors  of 

2.  63?  6.  175?  10.  264?  14.  440? 

3.  72?  7.  M7?  11.  200?  <      15.  500? 

4.  84?  8.  275?  12.  256?  16.  648? 

5.  96?  9.  325?  13.  250?  17.  900? 


90  INTERMEDIATE    ARITHMETIC. 

18.   What    is    the    greatest    common    divisor    of 
126  and  210? 

PROCESS.  Eesolve  126  and  210  into 

19£  _  0  v  *  v  Q  v  tf  their     prime     factors.        The 

i_,o  —  £  A  0  X  o  x  /'  e  ^    f    < 

01  A        •*  vx  *  \/  K  \/  •*  product  of  the  factors  common 

^iu  =  £  x  *>  X  o  X  ft  ,,,       .„   ,      ^u 

to  both  will  be  the  greatest 


2  X  3  X  7  r=  42,   J.MS.      common  divisor  required. 

NOTE.  —  This  process  and  that  for  finding  the  least  com- 
mon multiple  (Art.  54)  may  be  easily  explained  by  means 
of  objects. 

What  is  the  greatest  common  divisor  of 

19.  54  and  90?  23.  81   and   135? 

20.  72  and  96?  24.  63,   84,    and    126? 

21.  75  and  90?  25.  96,    144,    and    192? 

22.  84  and  108?  26.  128,    224,   and   320? 

Art.  48.  A  Divisor  of  a  number  is  a  number 
that  will  exactly  divide  it. 

A  Common  Divisor  of  two  or  more  num- 
bers is  a  number  that  will  exactly  divide  each 
of  them. 

The  Greatest  Common  Divisor  of  two 
or  more  numbers  is  the  greatest  number  that 
will  exactly  divide  each  of  them. 

Art.  49.  A  Prime  Number  is  one  that  has 
no  divisor  except  itself  and  one. 

A  Composite  Number  is  one  that  has  other 
divisors  besides  itself  and  one. 

Art.  50.  An  Even  Nnmber  is  exactly  divisi- 
ble by  2;  as,  2,  4,  6,  8,  10,  12,  etc. 


PROPERTIES    OF    NUMBERS.  91 

An  Odd  Number  is  not  exactly  divisible  by 
2;  as,  1,  3,  5,  7,  9,  11,  13,  etc. 

All  even  numbers  except  2  are  composite. 

Art.  51.  RULES.  —  1.  To  resolve  a  composite 
number  into  its  prime  factors,  Divide  it  by  any 
prime  divisor,  and  the  quotient  by  any  prime 
divisor,  and  so  continue  until  a  quotient  is  ob- 
tained which  is  a  prime  number.  The  several 
divisors  and  the  last  quotient  are  the  prime 
factors. 

2.  To  find  the  greatest  common  divisor  of  two 
or  more  numbers.  Resolve  the  given  numbers  into 
their  prime  factors,  and  select  the  factors  which 
are  common.  The  product  of  the  common  factors 
will  be  the  greatest  common  divisor. 

LESSON    II. 
Multiple  find  Least  Common  Multiple. 

Art.  52.  When  a  number  is  multiplied  by  an 
integer,  the  product  is  called  a  Multiple.  Thus, 
36,  or  12  X  3,  is  a  multiple  of  12. 

1.    What   number   is   a  multiple   of  3?    4?    5? 
7?    8?    10?    15?    20?    25?    30? 

2.  What  number  is  a  multiple  of  18?    24?     35? 
45?    44?    60?    100?    250? 

NOTE.  —  The  teacher  should  show  that  a  number  has  any 
number  of  multiples. 

3.  What   number    is    a    common    multiple    of    3 
and  4?     4  and  5?     6  and  8?     5  and  6  ? 


92  INTERMEDIATE    ARITHMETIC. 

4.  What    number    is    a    common    multiple    of    7 
and  5?     6  and   9?     3,  4,  and   6?     4,  8,  and  12? 

NOTE.  —  The  teacher  should   show  that  two  or  more  num- 
bers have  any  number  of  common  multiples. 

5.  What  is  the  least  common  multiple  of  3  and 
4?     5  and  6?     3,  6,  and  12?     2,  4,  and  8? 

6.  What  is  the  least   common   multiple  of  3,  5, 
and  10?     2,  5,  and  10?     2,  3,  5,  and  10? 

WBITTEN    EXERCISES. 

1.   What    is   the    least  common    multiple   of  12, 
18,   and   30? 

PROCESS.  Resolve    the 

-19  4\/d\/q  numbers      into 

I-j: £    X    £    X    ^  ,,       .  .  f 

18  —  2  v  *  v  *  their  prlme  fac" 

30       ,X3X£  tors,  and  select 

6()    ~L  X  6  X  ? all  the  different 


2X2X3X^X5  =  180,   L.  C.  M.         factors,  repeat- 
ing     each      as 

many  times  as  it  is  found  in  any  number.  The  factor  2  is 
found  twice  in  12;  the  factor  3,  twice  in  18;  and  the  fac- 
tor 5,  once  in  30.  The .  product  of  2X2X3X3X5  is 
the  least  common  multiple  required. 

What  is  the  least  common  multiple  of 

2.  12,  15,   and  20?  7.  24,    72,   18,   48? 

3.  21,  24,   and  42?  8.  15,   24,   18,    32? 

4.  32,  48,    and  80?  9.  75,    150,   300? 

5.  27,  54,    and  108?  10.  125,    250,    500? 

6.  24,  80,   and  120?  11.  $48,    $72,    $144? 

Art.  53.     A     Multiple     of    a    number    is     any 
number  which  it  will  exactly  divide. 


PROPERTIES    OF    NUMBERS.  93 

NOTE.  —  Every  number  is  an  exact  divisor  of  its  product 
by  an  integer. 

A  Common  Multiple  of  two  or  more  num- 
bers is  any  number  which  each  of  them  .will  ex- 
actly divide. 

The   Least   Common   Multiple   of  two   or 

more  numbers  is   the  least  number  which   each   of 
them  will  exactly  divide. 

Art.  54.  RULE.  —  To  find  the  least  common  mul- 
tiple of  two  or  more  numbers,  Resolve  the  numbers 
into  their  prime  factors,  and  then  select  all  the 
different  factors,  taking  each  the  highest  number  of 
times  it  is  found  in  any  number.  Multiply  the 
factors  thus  selected;  their  product  ivill  be  the 
least  common  multiple. 


QUESTIONS  FOR  REVIEW. 

What  is  meant  by  the  divisor  of  a  number?  When  is 
a  divisor  a  common  divisor?  Define  a  common  divisor. 
What  is  the  greatest  common  divisor  of  two  or  more 
numbers?  How  is  it  found? 

By  what  may  every  number  be  divided  ?  What  is  a 
prime  number?  A  composite  number?  What  is  an  even 
number?  An  odd  number? 

What  is  meant  by  the  factor  of  a  number?  A  prime 
factor?  A  composite  factor?  How  may  a  composite  num- 
ber be  resolved  into  prime  factors  ? 

What  is  a  multiple  of  a  number?  When  is  a  multiple 
a  common  multiple  ?  Define  a  common  multiple.  What 
is  the  least  common  multiple  of  two  or  more  numbers  ? 
Give  the  rule  for  finding  the  least  common  multiple. 
What  is  the  difference  between  a  divisor  and  a  multiple  f 


94  INTERMEDIATE    ARITHMETIC. 

SECTION     VXI. 

FRACTIONS. 


LESSON   I. 
The  Idea  of  a  Fraction   developed. 

1.  If   a    melon    be    cut    into    two    equal    pieces, 
what  part  of  the  melon  will  one  piece  be  ? 

2.  How  many  halves   in   a   melon?     How  many 
halves  in  any  thing  ? 

3.  If   a   melon  be    cut    into   four    equal    pieces, 
what  part  of  the   melon  will  one  piece  be?     Two 
pieces  ?     Three  pieces  ? 

4.  How  many  fourths  in  an  apple?     How  many 
fourths  in  any  thing  ? 

5.  Which  is  the  greater,  one  half  or  one  fourth 
of  an  apple?     How  many  fourths  equal  one  half? 


FRACTIONS.  95 

6.  If  a   cake   be   cut  into   three   equal   pieces, 
what  part  of  the  cake  will   one  piece   be  ? 

7.  How  many   thirds   in   a    cake?     How   many 
thirds  in  any  thing? 

8.  If    a    cake    be    cut    into    six    equal    pieces, 
what   part   of  the    cake    will    one   piece   be  ?     Two 
pieces?    Three  pieces?    Four  pieces?    Five  pieces? 

9.  How  many  sixths  in  any  thing  ? 

10.  Which    is    the    greater,    one    third    or    one 
sixth    of  a    cake  ?     How    many   sixths    equal    one 
third? 

11.  A  single  thing  is  a  unit.     How  many  halves 
in  a  unit  ?    How  many  thirds  ?    How  many  fourths  ? 
How  many  sixths  ? 

12.  What  is  meant  by  one  third? 

Ans.  One  third  is  one  of  the  three  equal  parts 
of  a  unit. 

13.  What  is  meant  by  two  thirds?     One  fourth? 
Three  fourths  ?     One  sixth  ?     Three  sixths  ? 

14.  Which  is  the  greater,  two  thirds  or  a  unit? 
Five  thirds   or  a  unit?     Three   fourths   or  a  unit? 
Four  fourths  or   a  unit  ? 

Art.  55.  Such  parts  of  a  unit  as  two  thirds,  three 
fourths,  five  sixths,  etc.,  are  called  Fractions. 

A  fraction  may  be  expressed  by  two  numbers, 
one  written  under  the  other,  with  a  horizontal 
line  between  them  ;  as,  f  ,  f ,  f  . 

The  number  below  the  line  denotes  the  number 
of  equal  parts  into  which  the  unit  is  divided.  It 
is  called  the  Denominator. 


96  INTERMEDIATE    ARITHMETIC. 

The  number  above  the  line  denotes  the  num- 
ber of  equal  parts  taken.  It  is  called  the  Nu- 
merator. 

Read  the  following  fractions,  and  tell,  in  each 
case,  into  how  many  equal  parts  the  unit  is  di- 
vided, and  how  many  parts  are  taken  : 

(15)  (16)  (17)  (18)  (19)  (20) 

4  7  1:2  T  7  25  23 


9_  _8_  13  13 

1  8 


_  __ 

13  15  1  8  3  0 


7  5  5  10  15  30 

g  g  1  1  i  a  2  o  5  o 

Write  the  following  fractions  in  figures  : 

(21)  (22)  (23) 

Two  fifths  ;        Seven  twelfths  ;  Twenty-four  fortieths  ; 

Seven  ninths  ;  Ten  thirteenths  ;  Thirty-five  fiftieths  ; 

Nine  tenths;    Forty  fiftieths;  Twenty-two  twelfths; 

Ten  ninths.      Twenty  seventeenths.  Forty  fifty-fifths. 


DEFINITIONS, 

Art,  56.  A  Fraction  is  one  or  more  of  the 
equal  parts  of  a  unit. 

Art.  57.  A  fraction  is  expressed  by  two  num- 
bers, called  the  Numerator  and  the  Denom- 
inator. 

The  Denominator  of  a  fraction  denotes  the 
number  of  equal  parts  into  which  the  unit  is 
divided. 

The  Numerator  of  a  fraction  denotes  the 
number  of  equal  parts  taken. 


97 
id  the 


FRACTIONS. 

The  numerator  and  denominator  are 
Terms   of  a  fraction. 

LESSON   II. 


Integers   and  Mixed  Numbers  reduced  to 
Fractions. 

1.  How    many    thirds 
in  an  apple  ?     How  many 
thirds  in   2   apples  ? 

2.  How  many   fourths 
in  3  pears? 

SOLUTION. — In  1  pear  there 
are  4  fourths,  and  in  2  pears 

there   are   twice  4  fourths,   which    are   8    fourths.     There 
are  8  fourths  in  2  pears? 

3.  How  many   sixths   in    3    oranges  ?      In   five 
oranges  ?     6    oranges  ?     8   oranges  ? 

4.  How  many  fifths  in  3  ?    5?     8?     10? 

5.  How  many  eighths  in  4  ?     6?    8?     10? 

6.  How   many 
halves  in  2  and  1 
half  oranges  ? 


SOLUTION.  —  In  2 
oranges  there  are 
twice  2  halves,  which 
are  4  halves,  and  4 
halves  arid  1  half  are 
5  halves.  There  are  5  halves  in  2  and  1  half  oranges. 

7.  How  many  fourths  in  2  and  3  fourths  ? 

8.  How   many   thirds   in    5    and    2    thirds  ?     8 
and  1   third?     7   and   2   thirds? 

I.  A.— 7. 


98  INTERMEDIATE     ARITHMETIC. 

9.   How   many    sixths    in   5    and    2    sixths  ?     8 
and   3   sixths  ?     12   and   5    sixths  ? 

10.  How   many    tenths    in    6    and    3    tenths?     7 
and   5   tenths  ?     8   and   7    tenths  ? 

11.  Read   6f;    33';    45|  ;    25}^;    50T^;    66,V 

12.  How  many   fifths   in   6f  ?     8|  ?     12|  ? 

WRITTEN     EXERCISES. 

13.  Reduce   157   to   ninths.     157?    to   ninths. 

PROCESS.  PROCESS. 

157  1575 

9  9 


1413  1413 

-y-,   An*.  7 


1420       . 
-p   An*. 

14.  Reduce  96  to  eighths.     96|  to  eighths. 

15.  Reduce  35  5  \  to  twelfths.     46,f  to  ninths. 

16.  Reduce  73j6f   to  elevenths.    63f  to  sevenths. 

17.  Reduce  53.V^  to  a  fraction. 
SUGGESTION.  —  Reduce  the  mixed  number  to  twentieths. 

18.  Reduce  33r4g   to  a  fraction. 
Reduce  to  a  fraction  : 

19.  85 &  22.    236|  25.    48£J- 

20.  36{i  23.      49^  26.    69 /fo 

21.  48^  24.      75^  27.    93,|>0 

Art.  58.    A  Mixed  Number  is  an  integer  and 
a  fraction  united;    as,  5£-,--16f, 


FRACTIONS.  99 

Art.  59.  RULES. — 1.  To  reduce  an  integer  to 
a  fraction,  Multiply  the  integer  ?»v  the  g'.ven  de- 
nominator, and  write  the  denominator  under  the 
product. 

2.   To  reduce    a    mixed    number   to    a    fraction, 

Multiply  the    integer    by    the    denominator    of    the 

fraction,  to   the   product    add    the  numerator,   and 

write  the  denominator  under   the   result. 

LESSON    III. 

Fractions   reduced  to  Integers   or  Mixed 
Numbers. 

1.  How     many 
pears    in    six    half- 
pears  ?     In  7  half- 
pears  ? 

2.  How     many 
days    in     10     half- 
days  ?     In  11  half- 
days  ? 

SOLUTION.  —  In   11  half-days  there   are  as  many 
as    two    half-days    are    contained    times    in    11    half-days, 
which  is  5J  times.     There  are  5J  days  in  11  half-days. 

3.  How   many   pints   in    14   half-pints  ?     In  17 
half-pints  ?     In  21  half-pints  ? 

4.  How   many   yards   in  18   thirds   of  a   yard  ? 
In  19  thirds  of  a  yard?     In  22  thirds  of  a  yard? 

5.  How  many  weeks  in  28  sevenths  of  a  week? 
30  sevenths  of  a  week  ? 


100  INTERMEDIATE    ARITHMETIC. 

6.  A   mason    was   17    half-days    in    building    a 
wall :    how  many  days  did  he  work  ? 

7.  A   boy   earned    25    fourths    of    a   dollar   by 
selling  papers  :    how  many  dollars  did  he  earn  ? 

8.  A  man  walked  25   eighths   of  a  mile  in  an 
how  many  miles  did  he  walk  ? 

9.  How  many  ones  in  4/  ?     5/  ?     y  ?     **  ? 

10.  How  many  ones  in  fg?     J-g?     -JJ?     93  j 

WBITTEN    EXERCISES. 

11.  Reduce    lf-£   to   a  mixed  number. 

PROCESS:    y/  —  177  -j- 15  =  11  Jf,  ^rcs. 

12.  Reduce  2r°/  to  a  mixed  number. 
Reduce  to  an  integer  or  mixed  number: 


13. 
14. 
15. 
16. 

94 
1  5 
1  0  5 

i  2 

307 
20 
360 
60 

17. 
18. 
19. 
20. 

7  5 

21. 

22. 
23. 
24. 

322 

25. 
26. 
27. 

28. 

744 
3  5 

3  1  5 

24 

3  1  2 

3  0 

1  6  0 

1  6 
42  1 

21 
350 

3  5 

630 
22 

796 
4  5 

80 
•220 

35 

504 
"4  1 

20 

Art.  60.    A  Proper  Fraction  is  one  whose 
numerator     is     less     than     its     denominator ;     as, 


An  Improper  Fraction  is  one  whose  nu- 
merator is  equal  to  or  greater  than  the  denom- 
inator; as,  f,  f,  f. 

The  value  of  a  proper  fraction  is  less  than  one;  and  the 
value  of  an  improper  fraction  is  equal  to  or  greater  than  one 


FRACTIONS. 


101 


Art.  61.  RULE. — To  reduce  an  improper  frac- 
tion to  an  integer  or  mixed  number,  Divide  Ike 
numerator  of  the  fraction'  by  the  denominator. 


LESSON    IV. 
Fractions   rediiced   to   Zsower  Terms. 

1.  How  many  half-inches 
in  2  fourths  of  an  inch  ?     In 
4  fourths  of  an  inch? 

2.  How   many    thirds    of 
an  inch  in   2   sixths  ?     In  4 
sixths  ?     In   6   sixths  ? 

3.  How  many  fourths   in 
6  eighths  ? 

SOLUTION.  —  In   2   eighths  there  is  1  fourth,  and  in  G 

eighths  there  are  as  many  fourths  as  2  eighths  are  con- 

tained times  in  6  eighths,  which  is  3.  There  are  3  fourths 
in  6  eighths. 

4.  How  many  fifths    in  2   tenths  ?     4   tenths  ? 
6  tenths?     8  tenths?     12  tenths? 

5.  How   many   thirds    in    6    fifteenths  ?      9    fif- 
teenths?    12  fifteenths?     18  fifteenths? 

6.  How  many  sevenths  in    6 

7.  How  many  eighths  in 

8.  Reduce  ^  ,  T8g  , 

9.  Reduce   J<  ,  ||,  and  A§  each  to  sevenths. 

10.  Reduce   if,  f!J,  and  ±j  each  to  eighths. 

11.  Reduce  -Jg,  |§,  and  f|  each  to   sixths. 

12.  Reduce        ?     5     an(j         each  to  tenths. 


2r 


-||?     ||? 


1  g  each  to  fourths. 


102  INTERMEDIATE     ARITHMETIC. 


WRITTEN    EXERCISES. 

13.   Reduce  |f  to  its  towest  terms. 

PKOCESS.  Reduce  ff  to  f  J  by  cli- 

63-^-3       21 7        3  viding   both   terms   by    3; 

"^"q  =  9S 7  ='  I'    Am'  DeXt    reduce     it     to     f     b>r 

OT: -7— O  £o /  db  ,.     .,.  1         .  t         «• 

dividing  both  terms  by  v  ; 

^,63 —  21  3  |  can  not   be    reduced   to 

84  —  21       4  lower  terms,  and,  hence,  is 

in  its  lowest  terms.  Or,  re- 
duce ff  to  f  by  dividing  both  terms  by  21,  the  greatest 
number  which  will  exactly  divide  each  term. 

NOTE.  —  The  teacher  should  show  that  the  value  of  a  frac- 
tion is  not  changed  by  dividing  both  of  its  terms  by  the  same 
number. 

Reduce  to  lowest  terms  — 

1  J.    64         1Q    72         99    105        9ft    231 
l-±.   JfJ        10.   T08-        L£.   ,T5        ^D.   3g3 

1^    72        1Q    32        9Q    13         97    144 
-   '   T44"        *  •'   25^        '***•   14^        ^'*   288 

1ft    96        90    96        94   225        9Ql2l 
J-O.   j?4        ^U.   T^a       ^-t.   315        ^O.   ||  J 

17    56        91     84        9?;    182        OQ   360 
*••   15  0        -jl*   196        4P«   i-g(j        w-   480 

Art.  62.  When  a  fraction  is  reduced  to  an 
equivalent  fraction  with  smaller  terms,  it  is  re- 
duced to  lower  terms. 

A  fraction  is  in  its  Lowest  Terms  when  no 
integer  except  1  will  exactly  divide  both  numer- 
ator and  denominator. 

Art.  63.  PRINCIPLE.  —  The  division  of  both  terms 
°f  a  fraction  by  the  same  number  does  not  change 
its  value. 


FRACTIONS.  103 

Art.  64.  RULE.  —  To  reduce  a  fraction  to  its 
lowest  terms,  Divide  both  terms  of  the  fraction  by 
any  common  divisor;  then  divide  both  terms  of  the 
resulting  fraction  by  any  common  divisor',  and  so 
on,  until  the  terms  of  the  resulting  fraction  have 
no  common  divisor  except  1.  Or : 

Divide  both  terms  of  the  fraction  by  their  great- 
est common  divisor. 

LESSON    V. 
Fractions   reduced  to  Higher   Terms. 

1.  How  many 
fourths  of  an  or- 
ange   in   1    half? 
In   2   halves  ? 

2.  How  many 

eighths  of  an  orange  in  1  fourth?     In  3  fourths? 

SOLUTION.  —  In  one  fourth  there  are  2  eighths,  and 
in  3  fourths  there  are  3  times  2  eighths,  which  are  6 
eighths.  There  are  6  eighths  in  3  fourths. 

3.  How  many  ninths  in  1  third  ?     In  2  thirds  ? 
3  thirds?     4  thirds? 

4.  How  many  tenths  in  §  ?    f  ?    f  ? 

5.  How  many  twelfths   in  §  ?    j  ?    |  ? 

6.  Change  f  and  g  each  to  twelfths. 

7.  Change  f ,   g ,  and  f   each  to  eighteenths. 

8.  Change  g,  |,  and  -J-J-  each  to  twenty-fourths. 

9.  Change  f ,  T7a ,  and  Y\  each  to  thirtieths. 
10.    Change   |,  [-J,  and  \  each  to  twenty-eighths. 


104  INTERMEDIATE    ARITHMETIC. 

WRITTEN    EXERCISES. 

11.  Change  |J  to  seventieths. 

PROCESS.  One  thirty-fifth  is   as  many  sev- 

70  -i-  35  =  9  entieths  as  35  is  contained  times  in 

70,  which  is  2  times,  and  17  thirty  - 

17  X  2  _  34^   £ns^        fifths    are    17    tjmes    2    seventieths, 
35  X  2       70  which    are   34   seventieths.     This   is 

the  same  as  multiplying  both  terms 
by  the  quotient  of  70  divided  by  35. 

12.  Change   J|  to  ninety-sixths. 

13.  Change   \\  and  fj  each  to  eighty-fourths. 

14.  Change  -^ ,   Af,    and    |g    each    to    seventy- 
seconds. 

15.  Reduce  f,  |,  and   }\  to  equivalent  fractions 
having  a  common  denominator. 

PROCESS. 

Change  the  fraction  to  twenty-fourths. 

~&  '  ^>          T2"*  time  •     5 20.        7    21.        11    22 

til  lib  .     TT    —   TTA  »       IT   —   "^t:  >       T7?"   —   ^d  • 

«,     II,     li- 

Reduce  to  a  common  denominator: 

1^35         7  1Q37         9  99         2         3         411 

LD-     4       6      T^  iy'     4       6       T5  ^3'       5      TO     T5     '30 

17.  §  -I     g          20.  f    I    T\         23.    I    ji   Jl  li 

18«    I     T7(5    T85  21-     i     T^     -\  24«     T^     H     M     to 

Art.  65.  When  a  fraction  is  changed  to  an 
equivalent  fraction  with  greater  terms,  it  is  reduced 
to  Higher  Terms. 

Several  fractions  having  the  same  denominator, 
are  said  to  have  a  Common  Denominator. 


FRACTIONS.  105 

Art.  66.  PRINCIPLE. — The  multiplication  of  both 
terms  of  a  fraction  by  the  same  number  does  not 
change  its  value. 

Art.  67.  RULES.  —  1.  To  reduce  a  fraction  to 
higher  terms,  Divide  the  given  denominator  by  the 
denominator  of  the  fraction,  and  multiply  both 
terms  by  the  quotient. 

2.  To  reduce  fractions  to  a  common  denom- 
inator, Divide  the  least  common  multiple  of  the 
denominators  by  the  denominator  of  each  fraction, 
and  multiply  both  terms  by  the  quotient. 

LESSON   VI. 
Addition  of  Fractions. 

1.  A  boy  gave  1  fourth  of  a  pine-apple  to  his 
brother,  1  fourth   to   his    sister,  and  1  fourth  to   a 
playmate :   what  part  of  it  did  he  give  away  ? 

How  many  fourths  are  \  +  J-  +  \  ? 

2.  A   grocer   sold   1  eighth   of  a   cheese    to  one 
customer,   2   eighths   to    another,  and   3    eighths   to 
another:   what  part  of  it  did  he  sell? 

How  much  is  £  +  §  +  f  ? 

3.  How   many   sixths   in    \ ,    § ,  and    f  ?      | ,  | , 
and  i?     §,  I,  and   |? 

4.  A    boy    gave    1    half    of    his    money   for    a 
knife,  and  1   third   of  it   for  a  ball :   what  part   of 
his  money  did  he  spend? 

SUGGESTION.  — Change  \  and   \  to  sixths. 


106  INTERMEDIATE     ARITHMETIC. 

5.  How  many  tenths  in  \   and   |  ?      |   and   T3^  ? 

6.  How  many  twelfths  in  J  and  \  ?      |  and   f  ? 

7.  How  many  eighths   in    \  and  |  ?     f  and  f  ? 

8.  How     many     fifteenths     in     J     and  ;l  ?      | 
and    f?      J   and  f?      f   and  f  ? 

9.  How     many    twentieths     in     \    and  f  ?      f 
and    -?  ?      |  and  f  ?      f   and  |? 

10.  How     many     twenty-fourths     in     .^,    -| ,     i, 
and   £  ?      In   J  ,   f ,   f ,  and  |  ? 

WRITTEN     EXERCISES. 

11.  What  is  the  sum  of   r53 ,  -,^  ,  ^ ,  ^  ? 
PKOCESS:    A  +  A  +  &  f  /,  -  f|  -  2TV,  J^. 

12.  What  is  the  sum  of    >f,    ||,    /T ,    and    \\1 

13.  What  is  the  sum  of  -JJ,    ||,    |g,    and    -fg? 

14.  What  is  the  sum  of   g ,    | ,   and   ^  ? 

PROCESS.  Change  the  fractions  to  twenty- 

5 .  _|_    7  _^_   s    —  fourths  ;    add   the  numerators    of 

20    i    21   i    i_o  _5i  tne  new  factions;  and  reduce  the 

resulting  improper  fraction  to  a 
Ji  ==  ^A  —  2-J-,  -4««.          mixed  number. 

15.  What  is  the  sum  of  | ,   g  ,  and   T\  ? 

16.  Add  |,  I,    and    | .' 

17.  Add  J,  /,,  and  Ji. 

18.  Add  f,  T30,  and   Jf! 

19.  Add  i,  f,    and  1J. 

20.  Add  §,  T3ff,    J^and   ja. 

21.  Add  I,  I,    f^  and  -JJ. 

22.  Add  «,,  ^,   |??  and  fj. 

23.  Add  ,,  --,        ,  and  |. 


FRACTIONS.  107 


24.   What  is  the  sum  of  16f  ,  18f,  and  37*? 


First  add  the  fractions   and  then  the  in- 
A          tegers.      f  =  A,    f  =  &,   \  =  A-      A  + 
2      ™          *  +  A-tt  =  ltt.     Write  the  }i  under 
~     —  —          the  fractions  and  add  the  1  with  the  integers. 


25.  Add  451,  67|,  and  62|  . 

26.  Add  37|,  18f,  33J  ,  and  25r72-. 

27.  Add  30  J,  66§,  84f,  and  133-J  . 

28.  Add  75i  ,  108,  160|,  and  207. 

Art.  68.  RULES.  —  To  add  fractions,  Reduce  the 
fractions  to  a  common  denominator,  add  the  nu- 
merators of  the  new  fractions,  and  under  the  sum 
write  the  common  denominator. 

2.    To   add  mixed   numbers,    Add    the  fractions 
and  the  integers  separately^  and  combine  the  results. 

LESSON    VII. 
Subtraction   of  Fractions. 

1.  Albert  had   2    thirds   of  an    orange   and   he 
gave  1  third  to   his    sister  :    how   many  thirds    had 
he  left? 

How  much  is  f  less  |  ?      |  less  |  ? 

2.  Charles    bought    3    fourths    of   a    pound    of 
raisins,  and  then  gave  1  fourth  of  a  pound  to  his 
playmate  :   what  part  of  a  pound  had  he  left  ? 

How  much  is  |  less  \  ?      J  less  \  ? 

3.  A  farmer  bought  |  of  a  bushel  of  flax-seed, 


108        INTERMEDIATE:   ARITHMETIC. 

and  sold  ^  of  a  bushel  to  a  neighbor :    what  part 
of  a  bushel  had  he  left? 

SUGGESTION.  —  Change    f   and   ^   to   sixths. 

4.  How  much  is  |  less  ^  ?     f  less  |  ? 

5.  How  much  is  |  less  |  ?     J  less   ~  ? 

6.  How  much  is  y7,  less  |  ?     'T7s  less   i  ? 

7.  How  much  is  T90  less  f  ?      ,9a  less  |  ? 

8.  How  much  is  }g  less  |?     j|  less  f  ? 

9.  How  much  is  J  less  <J  ?     r8^  less  T35  ? 

WRITTEN    EXERCISES. 

10.  From   1J-  take   ^Y 

PROCESS  :    ||-  —  •£?  —  ~h  —  i  >  ^^s- 

11.  From  p  take   J-J. 

12.  From    /^  take  ^o  . 

13.  From   J-J-   take  g. 

PROCESS  :   H  -  I  =  If  ~  if  =  A,  ^^- 

14.  Take  f  from    ^  ;     f  from  | . 

15.  Take  T75-  from    ^  ;     T70-  from   j  | . 

16.  Subtract  J|   from   |g  ;     !-§  from   :•?-. 

17.  Subtract  T5g  from  T7.>  ;     j7^  from  7. 

18.  Subtract  y\  from   28f  ;     ij-  from   J|. 

19.  Subtract  T52-  from  T\  ;     }£  from  {^. 

20.  From  33|  take  18|. 

First  subtract  the  fractions  and  then  the 
integers.     Since  T\  is  greater  than  -£%,  add 
33J        T\  }f  to  T42,  making  -}-§,  and  then  take  the  ^ 

18}  T92-  from  |f,  writing  T7^  under  the  fractions, 
and  adding  1  to  the  8  units  before  subtract- 
ing the  integers. 


FRACTIONS.  109 

21.  Take  30*   from  66f  ;    45|  from  66f  . 

22.  Take  112|  from  145*  ;    90|  from  108' . 

23.  Subtract  250|  from  300;    105f  from  261-J . 

24.  Subtract  130|  from  241f  ;   166|  from  233*. 

Art.  69.  RULES.  —  1.  To  subtract  fractions,  Re- 
duce the  fractions  to  a  common  denominator,  sub- 
tract the  numerator  of  the  subtrahend  from  the 
numerator  of  the  minuend,  and  under  the  differ- 
ence write  the  common  denominator. 

2.  To  subtract  mixed  numbers,  First  subtract 
the  fractions  and  then  the  integers. 

LESSON   VIII. 

^Problems  involving  the  JLddition   and  Subtrac- 
tion of  Fractions. 

1.  A  boy  spent  \  of  his  money  for  a  sled,  and 
§   of   it   for   a   pair  of    skates :    what    part    of   his 
money  had  he  left? 

2.  John   bought   a   knife   for   |    of  a   dollar   and 
a  ball   for   \   of  a   dollar,   and   then    sold   both   of 
them   for    ?    Of   a   dollar.     What   part   of  a   dollar 
did  he   gain  ? 

3.  Jane   having  J  of  a   quart   of  plums,  gave   ] 
of  a  quart  to  her  brother  and  \  of  a  quart  to  her 
sister  :    how  much  had  she  left  ? 

4.  A  farmer  bought   \\    bushels    of  clover-seed, 
and  then  sold  §   of  a  bushel   to   one   neighbor  and 
|  of  a  bushel  to  another :    what  part   of  a  bushel 
had  he  left? 


110  INTERMEDIATE    ARITHMETIC. 

5.  A    student    spends    §    of   his  time   in    study, 
y1^  of  it   in  labor,  and  J   of  it  in  sleep  :   what  part 
has  he  left? 

6.  One  sixth  of  a  pole  is  in  the   ground,   g   of 
it  in  water,  and  the  rest  in  the  air  :    what  part  is 
in  the  air  ? 

7.  A   man    bequeathed    •*    of   his    estate    to   his 
wife,    \    of   it   to    each    of   his   two    sons,    and   the 
rest  to  his   daughter  :   what  part  did   the   daughter 
receive? 

8.  A  man   did   \    of  a  piece  of  work   the   first 
day,   I   of  it   the   second   day,  and   then   completed 
it    the    third  :   what    part    of    the    work    did    he    do 
the  third  day  ? 

WRITTEN    EXERCISES. 


9.    From  the  sum  of  |  ,   §  ,  and  f  take 

10.  From   the    sum    of   |  and    f   take    their  dif- 
ference. 

11.  A  man  owning  \?z  of  a  vessel,  sold  \  and  ^ 
of  the  vessel  :    what  part  had  he  left  ? 

12.  A   farmer  bought   240^    acres    of  land,   and 
sold    90|    acres    and    75  J    acres:    how    many   acres 
had  he  left? 

13.  From   a  piece   of  broadcloth   containing  20  1 
yards,  a  merchant  sold  5J-  yards,  4*   yards,  and  8\ 
yards  :    how  many  yards  were  left  ? 

14.  A  man    earned    $56|    one    month   and   $70g 
the   next,    and   then    gave   $85-|    for   a  horse  :    how 
much  money  had  he  left  ? 


FRACTIONS.  Ill 

15.  From   the  sum    of  47  f    amd    33 J    take   their 
difference. 

16.  A  pedestrian  walked  T45   of  his  journey  the 
first  day,  y'V,-  of  it  the  next  day,  and   completed  it 
the   third   day :    what  part   of  the    journey   did   he 
travel  the  third  day  ? 


LESSON    IX. 
fractions  multiplied  by  Integers. 


1.  What     part     of     a 
cake    is    twice    2    eighths 
of  it  ?     3  times  2  eighths 
of  it  ? 

2.  A    father    gave    3 
fourths    of   an    orange    to 
each  of  4   children :    how 

many   fourths    did   they  all   receive? 

3.  How  much  is  4  times  f  ?     6  times  f  ? 

4.  If  a  boy  earn  2  thirds  of  a  dollar  in  a  day, 
how  much  will  he  earn  in  3  days  ? 

5.  -How  much  is  3  times  f  ?     5  times  f  ? 

6.  How  much  is  6  times  f  ?     9  times  f  ? 

7.  How  much  is  7  times  £  ?     8  times  J  ? 

8.  How  much  is  5  times  6|  ?     7  times  8^  ? 

SUGGESTION.  —  Multiply  the   integer  and  the  fraction   sep- 
arately, and  add  the  products. 

9.  How  much  is  3  times  6§  ?     8  times  7f  ? 

10.  How  much  is  6  times  4|  ?     9  times  8f .  V 

11.  How  much  is  8  times  6|  ?     8  times  7f  ? 


112 


INTERMEDIATE:  ARITHMETIC. 


Multiply : 

12.  V*  by     8. 

13.  /„  by  12. 

14.  i|  by  24. 


WRITTEN    EXERCISES. 


15.  fi  by  25. 

16.  11  by  16. 

17.  i?  by  33. 


18.  i*  by  16. 

19.  16}  by    4. 

20.  18}  by  12. 


Art.  70.  PRINCIPLE.  —  A  fraction  is  multiplied  by 
multiplying  its  numerator  or  dividing  its  denominator. 

Art.  71.  RULES.  —  1.  To  multiply  a  fraction  by 
an  integer,  Multiply  the  numerator  or  divide  the 
denominator. 

2.  To  multiply  a  mixed  number  by  an  integer, 
Multiply  the  fraction  and  the  integer  separately , 
and  add  the  products. 

.  LESSON   X. 
Fractional    'Parts  of  Integers. 

1.  If  6  pears  be  divided  equally  between  2  boys, 
what  part  of  the  whole  will  each  receive  ? 

What  is  \  of  6  pears?     \  of  10  pears? 

2.  A  father 
divided  5  mel- 
ons       equally 
between      two 
children:  what 
part     of     the 
whole  did  each 
receive  ? 

What    is    £ 
of   5    melons  ? 


FRACTIONS.  113 

SUGGESTION.  —  First  take  1   half  of  4  melons  and  then  1 
half  of  1  melon. 

3.  Charles   divided   12    plums    equally   between 
3   boys  :    what    part    of    the    whole    did    each    re- 
ceive? 

What  is   *    of  12  plums  ?     -J  of  13  plums  ? 

4.  What  is  -J  of  9  ?  -J  of  12  ?  '   of  16  ? 

5.  What  is  J  of  20  ?  i  of  28  ?  J  of  30  ? 

6.  What  is  J  of  25?  '   of  26?  1  of  37? 

7.  What  is  ft  of  24  ?  §  of  24  ? 

SOLUTION.  —  £  of  24  is  4,   and  £  of  24  is  5  times  4, 
which  is  20.    f  of  24  is  20. 

8.  What  is  ft  of  40  ?      ft  of  40  ?     ft  of  40  ? 

9.  What  is  |  of  63  ?      ?  of  64  ?     f  of  65  ? 

10.  What  is  $  of  45  ?      |  of  37  ?      |  of  58  ? 

11.  What  is       of  33?         of  58?  of  50? 


WRITTEN    EXERCISES. 

12.   What  is  f  of  659  ? 

PROCESS. 

8)659_  659 

octt 

Or: 


3  8)1977 

247  J,  Ans.  247J 

13.  What  is  f  of  191?  i  of  367? 

14.  What  is  |  of  508?  ft  of  243? 

15.  What  is  f  of  466?  T6f  of  4648? 

16.  What  is  /.2  of  906?  -Jg  of  6070? 

I.  A.— 8. 


114  INTERMEDIATE    ARITHMETIC. 

Integers  multiplied  by   Fractions. 
17.   Multiply  256  by  f . 

PROCESS.  |  is  3  times  -J-,  and  hence 

4)256  256  f  times  256   is  3  times  \  of 

64  Or.  256.     Or,    f    is    J    of  3,   and 

4)768  hence   f    times    256    is    |    of 

192,  Ans.  192  3   times  256. 


18.  48  by  ,V 
19.  65  by  3  . 
20.  59  by  f  . 
21.  87  by  I  . 

22.  163  by  ft  . 
23.  300  by  jf  . 
24.  257  by  'I. 
25.  305  by  *  |. 

26.  248  by  if. 
27.  406  by  ~\  • 
28.  856  by  if. 
29.  791  by  -11. 

30.   Multiply  324  by  16f . 

PROCESS. 

324 

First  multiply  by  the  integer  and  then 

by  the  fraction,  and  add  the  products. 
324 

216 


5400,    Ans. 

31.  48  by  16  J .  34.  246  by  12f  .  37.  108  by  56? . 

32.  72  by  18f .  35.  324  by  17  J .  38.  524  by  72| . 

33.  96  by  23| .  36.  406  by  33| .  39.  684  by  66- . 


Art.  72.  RULES.  —  1.  To  find  the  fractional  part 
of  an  integer,  or  to  multiply  an  integer  by  a 
fraction,  'Divide  the  integer  by  the  denominator  and 
multiply  the  quotient  by  the  numerator.  Or: 

Multiply  the  integer  by  the  numerator  and  divide 
the  product  by  the  denominator. 


FRACTIONS.  115 

2.  To  multiply  an  integer  by  a  mixed  number, 
Multiply  by  the  integer  and  the  fraction  separately, 
and  add  the  products. 

LESSON    XL 

Compound  fractions    reduced  to  Simple 
fractions. 


1.  If  each   third    of   a  pine-apple    be    cut  into 
2   equal  pieces,   what   part   of   the    pine-apple   will 
one  piece  be? 

What  is  A  of  -»  ? 

2.  A  boy  having  1  of  a  melon,  gave  -*-  of  it  to 
a  playmate :  what  part  of  the  melon  did  the  play- 
mate receive  ? 

What  is   *   of  \  ?     •'.  of  \  ? 

3.  What  is  \  of  i  ?     \  of  *  ?      <    of  J  ? 

4.  What  is   *   of  \  ?     I :  of  J  ?   .  *   of  £  ? 

5.  What  is  -'    of  1  ?     -J  of  4  ?     J  of  7J  ? 

6.  A   girl    having    |    of    an    orange,    divided   it 
equally  between  her  2  brothers :   what  part  of  the 
orange  did  each  receive  ? 

SUGGESTION.  —  Divide  each  fourth  into  2  equal  pieces,  and 
then  give  3  pieces  to  each. 

7.  What  is   *   of  -J  ?     J  of  f  ? 


116  INTERMKDIATE    ARITHMETIC. 

8.  What  is  -J  of  I  ?     i  of  f  ?     |  of  |  ? 

9.  What  is  £  of  §  ?     i  of  f  ?     f  of  f  ? 

SOLUTION.— J  of  f  is  fo,  and  f  of  f  is  3  times  ^, 
which  is  A.     foff^V 

10.  What  is   f  of  §  ?     |  of  §  ?     f  of  f  ? 

11.  What  is  |  of  f  ?     f  of  f  ?     |  of  T^  ? 

12.  What  is  J  of  12i  ?     ]   of  13^  ? 

SOLUTION.— J  of  13^  =  £  of  12  +  £  of  1£  or  f .    4  of 
12  is  4,  and  J  of  f  is  f  or  J.     Hence,  ^  of  13^  is  4J. 

13.  What  is  I  of  18-J  ?     J  of  21|  ?      |-   of  31  f  ? 

14.  What  is  4  of  22 1  ?     '  of  42  >-  ?      -J   of  46 f  ? 

15.  What  is  i  of  33'  ?     i  of  64J  ?     ^  of  62-J  ? 


\VRITTEN    EXEKCISES. 

16.    Reduce  |   of  |  to  a  simple  fraction. 

2     -  3        2X3         6         2 
PROCESS:    -  of  -  =  —  =  -  =  -,  An,. 

„        2,3        2X$        2 
Or''    3  °f  5  =  $>T5  =  5- 

NOTE.  —  The  examples  should  be  solved  by  both  methods. 
The  teacher  should  explain  the  process  of  cancellation. 


Reduce  to  a  s.imple  fraction  : 


17.  I  of  |  . 

21.   1   of  /r. 

25. 

18.  |  of  f  . 

22.   f   of  if. 

26. 

19.  f  of  /,. 

23.  A  of  T»s. 

27. 

20.  |  of  TV 

24.  T*3  of  |f  . 

28. 

f  of  £  of  f  . 
§  of  f  of  2J  . 
|  of  2J  of  |  . 
|  of  f  of  If  . 


FRACTIONS.  117 

Fractions   multiplied   by  Fractions. 

29.   Multiply  -f  by  f . 


PROCESS. 

4v/3      4X3       12 

3 

Since  f  is  f  of  one, 
J  times   |  =  f  of  once 

.£       nr    4-     r»f    -4       Avliipli 

5  ' 

N4 

5X4 

20 

5' 

om 

iala                                   TT£kn/->f» 

"*  4X5* 

Or  - 

4  N 

3 

4X3 

3           A                       4 
/I^j  o                        \ 

3       4X3 

3 

5' 

x  4 

5X4 

5'                  5> 

<4      5X4 

~5' 

29. 

1 

by 

§  • 

33.   J§ 

by 

8 
T3  ' 

37. 

2i. 

by   2J. 

30. 

4 

by 

2 

34.  |? 

by 

34 

3  0  ' 

38. 

"i 

by  3|. 

31. 

135 

V 

5 
U    • 

35.    | 

by 

9 
20  ' 

39. 

6J 

by  2". 

32. 

6 

G 

by 

T63-- 

36.  a»5 

by 

^- 

40. 

61 

by  121  . 

Art.  73.  A  Simple  Fraction  is  a  fraction  not 
united  with  an  integer  or  another  fraction ;  as,  | . 

A  Compound  Fraction  is  a  fraction  of  a 
fraction ;  as,  f  of  -| ;  f  of  3| . 

Art.  74.  RULE.  —  To  reduce  a  compound  frac- 
tion to  a  simple  fraction,  or  to  multiply  one  frac- 
tion by  another,  Multiply  the  numerators  together, 
and  also  the  denominators. 

NOTE. — The  process  may  often  be  shortened  by  canceling 
common  factors  before  multiplying. 

LESSON   XII. 
Fractions  diridecl  by  Integers. 

1.  A  father  divided  f  of  a  melon  equally  be- 
tween 3  boys:  what  part  of  the  melon  did  each 
receive  ? 


118  INTERMEDIATE    ARITHMETIC. 

SOLUTION.  —  If  3  boys  receive  f  of  a  melon,  each  will 
receive  J  of  f ,  which  is  ]-. 

2.  A  woman  divided  |   of  a  pound  of  crackers 
equally  between   3   poor  children:    what  part   of  a 
pound  did  each  receive  ? 

How  much  is  |  divided  by  3  ? 

3.  If  5   pounds   of  cheese  cost   f   of  a  dollar, 
what  will  1  pound  cost? 

4.  How  much  is  f  -=-  5  ?     f  -5-  5  ?     f  -5-  5  ? 

5.  If  6   men    can  build  |  of  a  wall   in   a  day, 
what  part  of  the  wall  can  one  man  build  ? 

6.  How  much  is  g  --  6  ?     f  ~  5  ?     J  -5-  9  ? 

7.  A   grocer   put   16 J    pounds  of  sugar  into  5 
equal  parcels :   how  much  sugar  was  put  into  each 
parcel  ? 

8.  Divide  16i  by  5.     12|  by  5.     18.J  by  5. 

9.  Divide  20|  by  3.     30f  by  4.     31  f  by  6. 

WRITTEN    EXERCISES. 

10.   Divide  T7^  by  3. 


PROCESS  : 
n^  . 

A  -5-  8  = 

7       •     Q- 

-J      Of       j?7J    = 

7 

7 

12    ' 

12X3 

36' 

11. 

A  by 

7. 

15. 

i 

4 
5 

by 

7. 

19. 

24$ 

by 

G. 

12. 

f    by 

12. 

16. 

i 
i 

6 

by 

8. 

20. 

29| 

by 

7. 

13. 

y72  by 

10. 

17. 

2 

0 
1 

by 

5. 

21. 

46  f 

by 

5. 

14. 

i83  by 

6. 

18. 

| 

i 

5 

by 

3. 

22. 

66| 

by 

8. 

Art.  75.    PRINCIPLE. — A  fraction    is    divided   by 


FRACTIONS.  119 

dividing  the   numerator   or   multiplying  the   denom- 
inator. 

Art.  76.  RULES. — 1.  To  divide  a  fraction  by  an 
integer,  Divide  the  numerator  or  multiply  the  de- 
nominator. 

2.  To  divide  a  mixed  number  by  an  integer, 
Divide  the  integral  part  and  then  the  fraction. 

LESSON    XIII. 
Integers   divided   by  Fractions. 

1.  How  many  times    is 
-J-    of   an   apple    contained 
in  2  apples  ?     f  of  an  ap- 
ple in  2  apples? 

2.  How  many  times  are 
\  of  a  yard  contained   in 
4  yards  ? 

SOLUTION.  —  4  yards  —  12  fourths,  and  3  fourths  are 
contained  in  12  fourths  4  times. 

3.  If   a  basket    holds  §  of  a  bushel,  how  many 
baskets  will  hold  4  bushels  ? 

4.  How   many   times   is    f    contained    in    4?     | 
in  4?     f  in  4? 

5.  If  |  of  a  yard   of  silk  will  trim   a   hat,  how 
many  hats  will  6  yards  trim  ? 

6.  How    many    times    is   |    contained    in    3  ?     f 
in  6  ?     f  in  6  ?     f  in  3  ? 

7.  Divide  12  by.f .     15  by  f .     20  by  f . 

8.  Divide     8  by  f .     10  by  | .     12  by  T6f. 


120  INTERMEDIATE    ARITHMETIC. 


WRITTEN    EXERCISES. 

9.   Divide  14  by  f . 

PROCESS. 

14  =  ™ .      ^-i-f  =  70-f.3  =  23J,  Am. 

NOTE.  —  Since  14  is  reduced  to  fifths  by  multiplying  it  by  5, 
the  process  may  be  shortened  by  omitting  the  denominators, 
thus :  13  -j-  | f  =  14  X  5  -f-  3  =  23}. 


10.  16  by  f  . 
11.  20  by  f  . 
12.  45  by  f. 

13.  60  by  J. 
14.  21  by  f. 
15.  42  by  f  . 

16.  30  by  2J. 
17.  40  by  3|. 
18.  16  by  5.J. 

Art.  77.  RULE.  —  To  divide  an  integer  by  a 
fraction,  Multiply  the  integer  by  the  denominator 
of  the  fraction,  and  divide  the  product  by  the  nu- 
merator. 

LESSON    XIV. 
Fractions   divided   by  fractions. 

1.  If  -|-   of  a  barrel  of  flour  will  supply  a  family 
one   month,   how   many   months  will   f    of  a  barrel 
last? 

2.  How   many   times    is   \    contained    in  f  ?     -J 

in    3  ? 
in   3  . 

3.  If  f    of  a   yard   of  cloth  will   make  a  vest, 
how  many  vests  will  f  of  a  yard  make? 

4.  How  many  times  f  in  |  ?     f  in  f  ? 

5.  If  a  pound  of  butter  cost  \  of  a  dollar,  how 
many  pounds  can  be  bought  for  f  of  a  dollar? 

SUGGESTION.  —  Change  \  to  eighths. 

6.  How  many  times  \  in  |  ?     J  in  |  ? 


3  > 

5. 

1 

by 

2 

3  • 

8. 

2J  % 

1 

i  ' 

6. 

1 

by 

•> 
f  * 

9. 

8*  by 

0 

H- 

7. 

s 

by 

f  • 

10. 

I  bJ 

FRACTIONS.  121 

7.  How  many  times  \  in  J  ?     -J  in  £  ?     £  in  J  ? 

8.  How  many  times   |   in  f  ?     -J  in  f  ?     f  in  f  ? 

WRITTEN    EXERCISES. 

9.  Divide  f  by  f  . 

PROCESS:    J  =  &.      | :  =  A.      T9?  -r-  A  =  §  =  H- 

2.  |   by 

3.  i   by 

4.  A  by 

Art.  78.  RULE.  —  To  divide  a  fraction  by  a  frac- 
tion, Reduce  the  fractions  to  a  common  denomina- 
tor, and  divide  the  numerator  of  the  dividend  by 
the  numerator  of  the  divisor. 

NOTE.  —  When  pupils  are  familiar  with  this  method^  they 
may  be  taught  to  divide  by  inverting  the  terms  of  the  divisor 
and  multiplying.  This  method  is  fully  explained  in  the  au- 
thor's Complete  Arithmetic. 

LESSON    XV. 
Numbers   Fractional    Tarts   of  Other  Members. 

1.  5  is  ^   of  what  number  ? 
SOLUTION. — 5  is  J-  of  3  times  5,  or  15. 

2.  7  is   J  of  what  number? 

3.  12  is  y1^  of  what  number  ? 

4.  12 J-  is  £  of  what  number? 

5.  16|  is  J  of  what  number  ? 

6.  10  is  |   of  what  number? 

SOLUTION.  —  If  10  is  f  of  a  number,  \  of  the  number 
is  J  of  10,  which  is  5 ;  if  5  is  J  of  a  number,  f  of  it  will 
be  3  times  5,  which  is  15. 


INTERMEDIATE    ARITHMETIC. 


7.  12  is   |  of  what  number? 

8.  25  is   I   of  what  number  ? 

9.  30  is   |   of  what  number  ? 

10.  33  is   §   of  what  number  ? 

11.  44  is  f   of  what  number? 

12.  A  man   spent  f   of  his  money  and  had  $21 
left:    how  much  money  had  he  at  first? 

13.  A  boy  gave  24  cents  for  a  slate,  which  was 
4  of  his  money:   how  much  money  did  he  have? 

14.  A   man    pays    $25    a   month   for   house-rent, 
which   is    i5*    of   his    monthly   salary :    what  is    his 
salary  ? 

15.  A  farmer  sold  a   cow  for  $45,  which  was  \ 
more  than  he  paid  for  her :   what  was  the  cost  of 
the  cow  ? 

1C.    A  man  sold  f  of  a  form  for  $1500 :   at  this 
rate,  what  was  the  value  of  the  farm  ? 

LESSON    XVI. 
Miscellaneous    "Problems. 

1.  Reduce  18|  to  an  improper  fraction. 

2.  Reduce   V^3  to  a  mixed  number. 

3.  Reduce  f  ,   | ,  and  |    to    a    common   denom- 
inator. 

4.  Add  i,  J,  },  and  T«3. 

5.  Add  28|,  40 >,  63f,  and  19  ,V 

6.  From  f  take  f ,     From  28f  take  16| . 

7.  Multiply  f  by  7.     13  by  f.     f  by  f . 

8.  Multiply  187|  by  15.     256  by  21|. 
0.   Divide  12  by  | .     f  by  12.     f  by  f . 


FRACTIONS.  123 

10.  Divide  243f  by  11.     256  by  5'  . 

11.  §  +  |-  what?     I -|?     f  X~f?     I-f-f? 

12.  There   are   5280   feet  in  a  mile :   how  many 
feet  in  ?9n  of  a  mile  ? 

13.  A  vessel  is  worth   $6000,  and  the  cargo  is 
worth  |  as  much  as  the  vessel :   what  is  the  value 
of  the  cargo  ? 

14.  A  man  sold  §  of  his  farm  to    one  neighbor 
and  §  of  it  to  another :  what  part  of  the  farm  has 
he  left? 

15.  A  man  owning  |  of  a  factory,  sold  §  of  his 
share :  what  part  of  the  factory  did  he  sell  ?     What 
part  does  he  still  own  ? 

16.  There    are    16^    feet   in    a   rod :    how   many 
feet  in  66  rods  ? 

17.  There   are    5^   yards    in    a   rod:    how   many 
rods  in  66  yards? 

18.  At  $6i  a  barrel,  how  many  barrels  of  flour 
can  be  bought  for  $150  ? 

19.  If  4  of  a  ship  is  worth  $12000,  what  is  the 
whole  ship  worth  ? 

20.  A   man    owning   f   of    an    estate,    sells  §  of 
his  share  for  $2400 :    at  this   rate,  what  would  be 
the  value  of  the  estate? 

21.  A   farmer   had    two    fields    of   wheat.      The 
first    yielded    840    bushels,    which    was    -f$    of    the 
amount  yielded  by   the    second  field.     How   many 
bushels  did  the   second  field  yield? 

22.  A   man   owning  |  of   a   ship,   sells  |   of  his 
share  for  $4400 :   at   this    rate,  what   is   the   value 
of  the  ship  ? 


124  INTERMEDIATE    ARITHMETIC. 

23.  The  value  of  a  certain  ship  is  $9760,  and 
§  of  the  value  of  the  ship  is  f  of  the  value  of  the 
cargo :  what  is  the  value  of  the  cargo  ? 


QUESTIONS  FOR  KEVIEW. 

What  is  a  fraction  ?  What  does  the  denominator  de- 
note? The  numerator?  What  are  the  terms  of  a  frac- 
tion? 

What  is  a  mixed  number?  What  is  meant  by  18}? 
Arts.  18  +  J .  When  is  a  fraction  called  proper  ?  When 
improper?  When  is  the  value  of  an  improper  fraction 
equal  to  1? 

How  is  an  integer  reduced  to  a  fraction?  How  is  a 
mixed  number  reduced  to  a  fraction?  What  kind  of  a 
fraction  is  the  result?  Give  examples.  How  is  an  im- 
proper fraction  reduced  to  a  whole  or  mixed  number? 
Give  examples. 

How  is  a  fraction  reduced  to  lower  terms?  On  what 
principle  does  the  process  depend?  How  may  a  fraction 
be  reduced  to  its  lowest  terms  by  one  division?  How  is 
a  fraction  reduced  to  higher  terms?  State  the  principle. 

How  are  fractions  having  a  common  denominator  added 
or  subtracted?  When  fractions  have  different  denomina- 
tors, how  are  they  added  or  subtracted?  How  may  mixed 
numbers  be  added?  How  may  they  be  subtracted? 

In  what  two  ways  may  a  fraction  be  multiplied  by  an 
integer?  How  may  an  integer  be  multiplied  by  a  frac- 
tion? Give  the  rule  for  multiplying  a  fraction  by  a  frac- 
tion. What  is  a  simple  fraction?  What  is  a  compound 
fraction  ?  How  is  a  compound  fraction  reduced  to  a  simple 
fraction  ? 

In  what  two  ways  may  a  fraction  be  divided  by  an 
integer?  How  may  an  integer  be  divided  by  a  fraction? 
How  may  a  fraction  be  divided  by  a  fraction? 


UNITED    STATES    MONEY. 


125 


SECTION     VIII. 

STATUS  MON&T. 


LESSON   I. 

Art.  79.  United  States  Money  is  the  legal 
currency  of  the  United  States.  It  is  also  called 
Federal  Money. 

The  denominations  are  mills,  cents,  dimes, 
and  dollars. 


126  INTERMEDIATE     ARITHMETIC. 

Table. 

10   mills    (m.)  .     .  are   1    cent,     .     .  c.  or  ct. 

10   cents    .     .    ..     .  are   1   dime,   .     .  d. 

10   dimes  ....  are   1   dollar,       .  §. 

$1  =  10  d.  =  100  c.  =  1000m. 

NOTES.  —  1.  United  States  money  consists  of  Coin  and  Paper 
Money.  Coin  is  called  Specie  Currency,  or  Specie,  and  paper 
money  is  called  Paper  Currency. 

2.  The  principal  gold  coins  are  the  eagle  ($10),  half-eagle 
($5),  quarter-eagle  ($2^),  three-dollar  piece,  and  dollar. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar, 
dime,  half-dime,  and  three-cent  piece. 

The  nickel  coins  are  the  live-cent  piece,  three-cent  piece, 
and  cent. 

The  copper  coins  (old)  are  the  two-cent  piece  and  the  cent. 

3.  Paper   money   consists    of   notes    issued    by    the   United 
States,  called  treasury  notes,  and  bank  notes    issued  by  banks. 

4.  Treasury  notes   of  a  value   less   than   $1,  as  fifty  cents, 
twenty-five  cents,  fifteen  cents,  ten  cents,  five  cents,  and  three 
cents,  are  called  Fractional  Currency. 

1.  How  many  mills   in  1  cent?     In   five   cents? 

4  cents  ?     7  cents  ?     9  cents  ? 

2.  How  many   cents   in   1   dime  ?     In  4   dimes  ? 

5  dimes?     8  dimes?     10  dimes? 

3.  How  many  dimes  in  1  dollar?     In  3  dollars? 

6  dollars  ?     4  dollars  ?     8  dollars  ? 

4.  How  many  cents  in  10  mills  ?     In   50  mills  ? 
40  mills?     60  mills?     80  mills? 

5.  How  many  dimes  in  40  cents  ?     In  90  cents  ? 
60  cents?     70  cents?     100  cents? 

6.  How    many    dollars    in     50    dimes?     In    70 
dimes?     60  dimes?     80  dimes?     100  dimes? 

7.  How  many  dimes  in  25  cents?     In  75  cents? 
15  cents?     35  cents?     95  cents? 


UNITED    STATES    MONEY.  127 

8.  How  many  cents  in  35  mills  ?     In  65  mills  ? 
25  mills?     75  mills?     95  mills? 

9.  How  many  cents  in  5  dimes  ?     In  15  dimes  ? 
45  dimes  ?     30  dimes  ? 

10.  How  many  dimes  in  15  cents?     In  95  cents? 
85  cents?     65  cents? 

11.  How  many  cents  in  1  dollar?     In  5  dollars? 
7  dollars?     9  dollars?     8  dollars? 

12.  How  many   dollars   in    200   cents  ?     In   500 
cents?     400  cents?     900  cents? 

WRITTEN     EXERCISES. 

Art.  80.  Accounts  are  kept  in  dollars,  cents,  and 
mills.  The  figures  denoting  dolhirs  are  preceded 
by  the  sign,  $'.  The  first  two  figures  at  the  right 
of  dollars  denote  cents,  and  the  third  figure  denotes 
mills.  The  figures  denoting  dollars  are  separated 
from  those  denoting  cents  by  a  period  ( . ),  called  a 
Separatrix.  Thus,  $45.307  is  read  45  dollars, 
30  cents,  7  mills. 

NOTE. — The  pupil  should  here  be  taught  that  the  first 
figure  at  the  right  of  the  separatrix  denotes  tenths  of  a  dollar; 
the  second,  hundredths ;  the  third,  thousandths.  lie  should  also 
be  taught  to  read  the  following  numbers  decimally,  and  to  write 
similar  numbers,  when  dictated,  decimally.  The  first  decimal 
period  may  thus  be  mastered. 

Copy  and  read  the  following  : 


(13) 

(14) 

(15) 

(16) 

$3.45 

$0.075 

$40.045 

$100. 

$3.506 

$0.005 

$15.15 

$405. 

$1.055 

$3.08 

$10.015 

$704.50 

$0.75 

$9.009 

$60.60 

$800.08 

128       INTERMEDIATE:   ARITHMETIC. 

17.  Write  in  figures  4  dollars  40  cents. 

18.  Write  12  dollars  33  cents  5  mills. 

19.  Write  60  dollars  6  cents  4  mills. 

20.  Write  75  cents  5  mills. 

21.  Write  30  cents;    40  cents  7  mills. 

22.  Write  300  dollars  3   cents  7  mills. 

23.  Write  500  dollars  5  mills. 

24.  Write  25  cents   7  mills ;    6  cents  1  mill. 

25.  Write  10  dollars  3  cents  8  mills. 

26.  Write  1000  dollars;    50  dollars  5  cents. 

27.  Write  25  dollars  1  cent  5  mills. 

28.  Write  500  dollars  3  mills. 

29.  Write  5  mills ;    5  cents  5  mills. 

30.  Write  60  dollars  60  cents  6  mills. 


LESSON   II. 
^Reduction   of  United  States   Money. 

MENTAL    AND    WRITTEN    EXERCISES. 

1.  How  many  cents  in  $15  ?         Ans.     1500^. 

2.  How  many  mills  in  $15?          Am.  15000m. 

3.  How  many  cents  in  $5.25  ?      Ans.       525c. 

4.  How  many  mills  in  $1.375?     Ans.     1375™. 

5.  How  many  mills  in  $.62 J?       Ans.       625m. 

6.  Reduce  $75  to  cents.     $108  to  cents. 

7.  Reduce  $125  to  cents.     $230  to  cents. 

8.  Reduce  $12.65  to  cents.     $5.60  to  cents. 

9.  Reduce  $1.08  to  cents.     $8.01  to  cents. 

10.  Reduce  $25  to  mills.     $40  to  mills. 

11.  Reduce  $.621  to  mills.     $3.12'   to  mills. 


UNITED    STATES    MONEY.  129 

12.  Reduce  $.375  to  mills.     $.105  to  mills. 

13.  Reduce  $4.50  to  mills.     $3.03  to  mills. 

14.  Reduce  $.45  to  mills.     $.05  to  mills. 

15.  Reduce  $102  to  cents.     $10  to  cents. 

16.  Reduce  $120  to  mills.     $45  to  mills. 

17.  Reduce  $.25  to  cents.     $.01  to  cents. 

18.  How  many  dollars  in  7500  cents? 

Am.  $75. 

19.  How  many  dollars  in  7550  cents? 

Ans.   $75.50. 

20.  How  many  dollars  in  3125  mills? 

Ans.  $3.125. 

21.  How  many  dollars  in  4000  mills? 

22.  Reduce  1507   cents  to  dollars. 

23.  Reduce  1001  cents  to  dollars. 

24.  Reduce  1500  mills  to  dollars. 

25.  Reduce  10250  mills  to  dollars. 

26.  Reduce  5000  cents  to  dollars. 

27.  Reduce  5000  mills  to  dollars. 

28.  Reduce  375  cents  to  dollars. 

29.  Reduce  375  mills  to  dollars. 

30.  Reduce  $4.50  to  mills. 

31.  Reduce  4500  mills  to  dollars. 

32.  Reduce  $10.10  to  cents. 

33.  Reduce  1010  mills  to   dollars. 

Art.  81.    RULES.  —  1.  To  reduce  dollars  to  cents, 
Annex  tivo  ciphers. 

2.  To    reduce     dollars    to    mills,    Annex    three 
ciphers. 

3.  To  reduce  cents  to  mills,  Annex  one  cipher. 


130  INTERMEDIATE    ARITHMETIC. 

4.  To    reduce    dollars    and    cents    to    cents,    or 
dollars,  cents,  and  mills   to  mills,  Remove  the  sep- 
aratrix  and  the  dollar  sign. 

5.  To    reduce    cents   to    dollars,   Place   the   sep- 
aratrix  before   the   second   right-hand  figure. 

6.  To    reduce    mills    to    dollars,   Place    the    sep- 
aratrix  before  the  third  right-hand  figure. 

NOTE.  —  Annexing  two  ciphers  is  multiplying  by  100,  and 
pointing   off  two   figures   from    the   right   is  dividing   by  100.  ^ 
'See  Arts.  37,  45.) 

LESSON    III. 
Addition  and  Subtraction. 

WRITTEN    EXERCISES. 

1.  What    is    the    sum   of   $50,    ,$16.50,    $3.333, 

and  $.87i-  ? 

PROCESS.  Write    the    several    numbers    to    be 

$50.  added  so  that   units   of  the   same    de- 

16.50  nomination    may    stand    in    the    same 

3.333  column,    and    then    add    as    in    simple 

.875  numbers.     The  dollar  sign  need  not  be 

$70.708,  Am.         written  but  once. 

2.  What  is  the  sum  of  $1.20,  $5,  $10.15,  $.85, 
and  $.62i? 

3.  What    is    the    sum    of    $9,    $12.50,    $4.37| , 
$40.08,   $6.33,   and   $25.? 

4.  Add  45.37',   $100.50,  $16.12 J,  $37,   $9.05, 
$.874,  $4.44,  and   $95. 


UNITKD    STATKS    MONKY.  131 

5.   From  $37.50  take  $5.62| -. 

PROCESS.  6.    From  $6.37*   take  $5.87f . 

$37.500  7.    From  $100.  take  $1.256. 

5.625  8.    From  $10.  take  $.10. 

$31.875,  Ans.  9.    A   man    sold    a    carriage    for 

$160.75,  a  horse  for  $125,  a  set 
of  harness  for  $26.37 '  ,  and  a  saddle  for  $15.62*  : 
what  was  the  amount  received? 

10.  A  grocer  buys  flour  at  $8.62i  a  barrel,  and 
sells  it  at  $10.  a  barrel:    what  is  his  gain? 

11.  A   merchant    paid    $32.50    for   a    barrel    of 
sugar,  and  sold  it  for  $35. :   how  much  did  he  gain  ? 

12.  A   laborer    earns    $17.50    a    week,    and    his 
expenses  are  $12.62  £  a  week:    how  much   can  he 
save  each  week  ? 

13.  A  man  bought  a  house  and  lot  for  $3506.75, 
and  sold  it  for  $4000 :   what  was  his  gain  ? 

14.  A   man    bought   a    carriage    for   $160,    paid 
$22.75  for  repairing  it,  and  then  sold  it  for  $180 : 
how  much  did  he  lose  ? 

15.  Mr.  Smith  bought  a  house  and  lot  for  $4500, 
and  paid    $40.50    for   a    front    fence,   $105.65   for 
painting    the    house,    $47.12    for   papering    several 
rooms,  and  $25  for  other  improvements :   what  will 
he  make  if  he  sell  the  property  for  $5000? 

Art.  82.  RULE.  —  To  add  or  subtract  sums  of 
money,  Write  units  of  the  same  denomination  in 
the  same  column,  add  or  subtract  as  in  simple  num- 
bei's,  and  separate  dollars  and  cents  by  a  period  and 
prefix  the  dollar  sign. 


132        INTERMEDIATE:  ARITHMETIC. 

LESSON   IV. 
Multiplication   and  Division. 

WRITTEN     EXERCISES. 

1.    What  will   9    cords    of  wood   cost,  at  $3.62  J 
a  cord  ? 

If  1  cord  of  wood  cost  $3.625,   nine 
cords   will   cost    9   times    $3.625,   which 


2.  What  will  16  barrels   of  flour  cost,  at  $7.50 
a  barrel  ? 

3.  What  will  40  yards  of  cloth   cost,  at  $1.12* 
a  yard.? 

4.  What  will  12|    tons    of  hay   cost,    at  $11.50 
a  ton? 

5.  At  18|    cents   a   dozen,  what  will   12    dozen 
of  eggs  cost? 

6.  If  a  boy  earn  $4.37^-  a  week,  how  much  will 
he  earn  in  20  weeks  ? 

7.  A  drover  sold   36   cows  at   $33.33'    a  head: 
how  much  did  he  receive  for  them  ? 

8.  What  will  90  bushels  of  wheat  cost  at  $1.62^ 
a  bushel? 

9.  If  9  cords   of  wood   cost  $32.62'-,  what  will 
1  cord  cost  ? 

PROCESS.  If  9    cords   cost   $32.625,   one   cord 

9)  $32.625  will  cost  $  of  $32.625,  which  is  $3.625. 

$3.625,   Am.         or  $3.62|. 


UNITED    STATES    MONEY.  133 

10.  If  12  pounds  of  sugar  cost  $2.16,  what  will 
1  pound  cost? 

11.  A  man  paid  $1687.50  for  45  acres  of  land: 
what  was  the  price  an  acre  ? 

12.  A  grocer  paid  $135  for  18  barrels  of  flour : 
what  was  the  cost  a  barrel  ? 

13.  A  man  earned  $91.  in  8  weeks :   how  much 
did  he  earn  a  week? 

14.  At  $.12j  a  dozen,  how  many  dozens  of  eggs 
can  be  bought  for  $5  ? 

PROCESS.  $5  =  5000  mills,  and  $.12J  = 

125 m. )  5000m.  ( 40,  Ans.        125  mills.     Hence,  $5.  -f-  $.12-| 

500  =5000  mills -f- 125  mills,  which 

0  is  40. 

NOTE.  —  When  both  divisor  and  dividend  are  denominate 
numbers,  they  must  be  reduced  to  the  same  denomination 
before  dividing. 

15.  At   $1.25    a  bushel,   how   many   bushels   of 
corn  can  be  bought  for  $75. 

16.  At  3^  cents  a  piece,  how  many  lemons  can 
be  bought  for  $7. 

17.  If  a  boy  earn  75  cents  a  day,  in  how  many 
days  will  he  earn  $24? 

18.  At   37i   cents   a   bushel,  how  many   bushels 
of  oats  can  be  bought  for  $57.75. 

19.  A   farmer   sold   35    pounds    of  butter  at  20 
cents  a  pound,  and  received  in  payment  muslin  at 
12*   cents  a  yard  :   how  many  yards  of  muslin  did 
he  receive  ? 

20.  A   farmer  exchanged  16    cows,  at   $27.50   a 


134  INTERMEDIATE    ARITHMETIC. 

head,  for  sheep  at  $5.50  a  head:   how  many  sheep 
did  he  receive? 

21.  How  many  lemons,  at  2^  cents  each,  can  be 
bought  for  20  oranges,  at  5  cents  each  ? 

22.  Multiply    $12.62*    by    15,    and    divide    the 
product  by  $2.525. 

23.  Multiply  $1.25   by  18,  and  divide  the  prod- 
uct by  $.62  J. 

24.  Multiply  $5.75  by  25,  and  divide  the  prod- 
uct by  $.57i-. 

Art.  82,  RULES.  —  1.  To  multiply  or  divide  sums 
of  money  by  an  abstract  number,  Multiply  or  divide 
as  in  simple  number  s,  separate  dollars  and  cents  in 
the  result  by  a  period,  and  prefix  the  dollar  sign. 

2.  To  divide  one  sum  of  money  by  another,  Re- 
duce both  numbers  to  the  same  denomination,  and 
divide  as  in  simple  numbers. 

LESSON   V. 
Miscellaneous  Written   "Problems. 

1.  What  is   the   sum   of  $13.45,   $9.87,   $100., 
$.87,   $1.40,  and  $14.? 

2.  From  $10  take  5  mills. 

3.  From  $500  take  500  cents. 

4.  Multiply  $15.33J  by  33. 

5.  Divide  $50  by  50  cents. 

6.  A  man's    income   tax   in  1868  was   $55.75, 
his  State  and  city  tax  $68.35,  and  his  other  taxes 
$7.50 :   what  was  the  amount  of  his  taxes  9 


UNITED    STATES    MONEY.  135 

7.  A  man   bought   a   house   and   lot  for  $5400, 
and,  after  expending  $1500  for  improvements,  sold 
the  property  for  $7500  :   how  much  did  he  gain  ? 

8.  What  will   60  pounds  of  butter  cost,  at  33| 
cents  a  pound  ? 

9.  What   is    the    cost    of   35    reams    of    paper, 
weighing  44  pounds  each,  at  18  cents  a  pound  ? 

10.  How  many  yards   of   carpeting,   at   $1.75 'a 
yard,  can  be  bought  for  $350? 

11.  A    fruit    dealer  makes    a    net    profit    of   20 
cents    on    each    bushel    of    apples    he    sells :    how 
many  bushels  must  he  sell  to  make  $80  ? 

12.  A  widow  is  to  receive  one  third  of  an  estate 
of  $12000,   and    the    remainder   is    to    be    divided 
equally  between  5  children :   what  is  the  share  of 
each  child  ? 

13.  A  fruit  dealer  sold  144  baskets   of  peaches 
for  $252 :   what  was  the  price  per  basket  ? 

14.  If  40  acres   of  land   cost  $1400,  how  many 
acres  can  be  bought  for  $1750  ?    . 

15.  A  man  sold  15    cords   of  wood,  at   $4.50  a 
cord,  and  received  in  payment  10  barrels  of  flour : 
what  did  the  flour  cost  him  a  barrel? 

16.  If  8  barrels  of  salt  cost  $36,  what  will  13 
barrels  cost? 

17.  A    grain    dealer    bought    15000    bushels    of 
wheat  at  $1.35  a  bushel,  and  sold  it  the  next  week 
for  $1.48  a  bushel :   what  was  his  gain  ? 

18.  A  workman  receives   $1.50   a  day,   and   his 
living  costs  him  $.75  a  day :   how  much  can  he  lay 
up  in  a  year,  if  he  work  310  days? 


136        INTERMEDIATE:   ARITHMETIC. 

19.  A  drover  bought  240  sheep  at  §4.50  a  head, 
drove  them   to  market  at  an   expense  of  $75,  and 
then  sold  them  at  $6.50  a  head  :   how  much  did  he 
make  ? 

20.  A  farmer  exchanged  40  pounds  of  butter,  at 
22  cents  a  pound,    and    8   dozen    of  eggs,    at  12^ 
cents  a  dozen,  for  cotton  cloth  at  10  cents  a  yard  : 
how  many  yards  of  cloth  did  he  receive  ? 

21.  A    farmer    exchanged     8     cows,    valued    at 
$37.50  a  head,  for  sheep  valued  at  $7.50  a  head: 
how  many  sheep  did  he  receive  ? 

22.  If  a  boy  pays  $2.50   a  hundred   for  papers, 
and  sells  them  at  5  cents  a  piece,  how  much  does 
he  make   on  100  papers  ? 

23.  A   farmer   sold,    one    year,   200   bushels    of 
wheat,    at    $1.80   a   bushel ;    500   bushels    of   corn, 
at  $1.15  a  bushel ;    65  bushels   of  potatoes,  at   80 
cents  a  bushel;    12  tons  of  hay,  at  $16.50  a  ton; 
and   225   pounds    of  butter,   at   20  cents  a  pound. 
What  was  the  amount  of  his  annual  product? 

24.  A  man  bought   250  bushels   of  coal,   at  15 
cents  a  bushel ;   7  cords  of  wood,  at  $5.50  a  cord ; 
18  bushels   of  potatoes,   at   $.90   a  bushel ;    and  9 
barrels    of  apples,   at   $2.75   a  barrel.     How  much 
did  he  pay  for  all? 

25.  A  bookseller  sold  12  geographies,  at  $1.75 ; 
20  readers,  at  $.85 ;    30  arithmetics,  at  $.65 ;    and 
45  spellers,  at  »$.30.     What  was  the  amount  of  the 
bill? 

26.  The  annual  expenses  of  a  man's  family  are 
as  follows :   provisions,  $350 ;   clothing,  $400  ;  fuel, 


UNITED    STATES    MONEY.  137 

$95;  books  and  periodicals,  $50;  house-rent,  $240; 
and  all  other  expenses,  $150.  If  he  receive  an 
annual  salary  of  $1500,  how  much  can  he  lay  up 
each  year? 

LESSON   VI. 


1.  COLUMBUS,  0.,  JUNE  10,  1869. 

MR.  CHARLES  WILSON 

Bought  of  JAMES  COOPER  &  Co.: 

13  Ibs.  Coffee,  @  30c.,  .........  $3.90 

4  Ibs.  Butter,  @  35c.,  .........     1.40 

10  Ibs.  Bk't  Flour,  @  6c.,  ..........  60 

12  Ibs.  Dr'd   Beef,  @  24c.,  .........     2.88 

25  Ibs.  Sugar,  @  18c.,  .........     4.50 

3  Ibs.  Starch,  @  20c.,  ..........  60 

$13.88 
Received  payment, 

JAMES  COOPER  &  Co. 


2.  CHICAGO,  JAN.  3,  1869. 

JOSEPH  MASON 

Bought  of  PETER  &  BROTHERS: 

27    yds.  Brussels  carpeting,  @  $2.60, 

23    yds.  Ingrain  "  @     1.75, 

8f  yds.  Oil  cloth,  @    1.20, 

32    yds.  Curtains,  @      .60, 


Received  payment, 

PETER  &  BROTHERS, 
Per  SMITH. 

What  is  the  amount  of  the  above  bill  ? 


138  INTERMEDIATE    ARITHMETIC. 

3.  NASHVILLE,  TENN.,  OCT.  8,  1868. 

SAMUEL  MILLS 

To  JONES,  SMITH  &  Co.,    Dr. 

To    7  yds.  Broadcloth,  @  $6.50, 

"     3J  yds.  Doeskin,  @  2.75, 

"      7f  yds.  Linen,  @  .90, 

"     2J  doz.  Handkerchiefs,  @  1.50, 

"  12J  yds.  Muslin,  @  .18, 

"      9  yds.       do.   bleached,  @  .33, 

"  12  yds.  Silk,  @  1.60, 

"  19  yds.  Binding,  @  .08, 


Received  payment, 

JONES,  SMITH  &  Co. 

What  is  the  amount  of  the  above  bill? 


4.  ST.  Louis,  MAY  23,  1869. 

HENRY  WILLIAMS 

*g,,g  Bought  of  ISAAC  CLAKKE: 

Mch.  10,  5  Pair    Calf  Boots,             @  $5.75,    .  . 

"       "  8  "      Laidies'  Gaiters,    @     3.10,   .  . 

"       "  7  "      Children's  Shoes,  @     1.75,   .  . 

Apr.      4,  8  "      Coarse   Boots,         @     2.75,   .  . 

"  6  "      Calf  Shoes,             @    3.25,   .  . 

"  7  "     Ladies'  Slippers,     @    1.20,   .  . 

May    23,  3  "      Calf  Boots,             @     5.75,   .  . 


Received  payment, 
What  is  the  amount  of  the  above  bill  ? 


UNITED    STATES    MONEY.  139 

5.  PITTSBURG,  PA.,  DEC.  15,  1868. 

ANDREW  WILSON 

Bought  of  SMITH  &  WARING: 

lobo. 

July    5,  7    gross  Shirt  Buttons,  @  $4.50, 

"       "  10     doz.    Linen   Napkins,  @  2.75, 

Aug.  12,  8       "      Pair  Kid   Gloves,  @  12.50, 

"      "  3}     "      Linen   Handk'fs,  @  6.75, 

"      "  4f     "      Shirt  Bosoms,  @  6.00, 

Dec.  15,  3f    "      Silk  Gloves,  @  9.00, 

"      "  8      "      Pair  Socks,  @.  5.50, 


$ 
What  is  the  amount  of  the  above  bill  ? 


6.  INDIANAPOLIS,  IND.,  AUG.  18,  1869. 

THOS.  M.  COCHRANE 

Bought  of  JONES,  DUNLAP  &  Co. : 

12     doz.   Scythes,          ©  $15,  

12}  doz.   Scy.  Snaths,  @  16.50, 

6     doz.   Kakes,            @  2.25, . 

5    doz.   Hoes,              @  5.75, 

8|  doz.   Whetstones,  @  1.50, 


Cr. 

June  20,     By  Cash, $75.00 

Aug.     1,     By    2}   doz.   Scythes   returned,    .     37.50 

$112.50 

$~ 
Received  payment, 

JONES,  DUNLAP  &  Co. 
What  is  the  amount  due? 


140  INTERMEDIATE     ARITHMETIC. 

7.  COLUMBUS,  O.,  JULY  1,  1869. 

SMITH  &  BELL 

^ggg  In  account  with  GEOEGE  STATIONER. 

Feb.  1,  To  2J  M.  Envelopes,  @  $5.75, 

"  "  "  1J  reams  Cap  Paper,  @  8.00, 

"  "  "  3  Blank  Books,  @  1.25, 

Mch.  9,  "  5  doz.  Pencils,  @  1.25, 

"  "  "  50  Ibs.  Wrapping  Paper,  @  .10, 

"  "  "  6  vols.  Dickens,  @  1.75, 

IT 
Or. 

June  20,  By  printing  1500  Circulars,  .  $5.50 
"  "  By  printing  Letter  Heads,  .  3.75 
"  25,  By  33  tokens  Press-work,  .  .  .50 


What  is  the  amount  due? 


DEFINITIONS, 

Art.  84.  A  Sill  of  Goods  is  a  written,  state- 
ment of  goods  sold,  with  the  price  of  each  article 
and  the  entire  cost.  It  also  gives  the  date  and 
place  of  the  sale,  and  the  names  of  the  buyer  and 
seller. 

A  bill  is  drawn  against  the  buyer,  or  Debtor, 
and  in  favor  of  the  seller,  or  Creditor. 

A  bill  is  receipted  by  writing  the  words  "Re- 
ceived payment"  at  the  bottom,  and  affixing  the 


UNITKD    STATES    MONKY.  141 

seller's  name.     A  bill  may  be  receipted  by  a  clerk, 
agent,  or  any  other  authorized  person,  as  in  bill  2. 

Art.  85.  When  sales  are  made  at  different  times, 
the  dates  may  be  written  at  the  left,  as  in  bills 
4,  5,  and  7. 

A  bill  presenting  a  debit  and  credit  account  be- 
tween the  parties,  may  be  written  and  receipted  as 
in  bill  6. 


QUESTIONS  FOR  REVIEW. 

What  is  United  States  money?  What  is  it  also  called? 
Of  what  does  United  States  money  consist?  What  is  each 
kind  of  money  called? 

What  are  the  principal  gold  coins  ?  Silver  coins  ?  Nick- 
el coins?  Copper  coins?  Name  the  two  kinds  of  paper 
money.  What  is  meant  by  "fractional  currency"? 

What  are  the  principal  denominations  of  United  States 
money?  Eepeat  the  table.  In  what  denominations  are 
accounts  kept?  What  use  is  made  of  the  dollar  sign? 
How  are  dollars  and  cents  separated?  Is  the  separatrix  a 
period  or  a  comma?  Where  is  the  figure  denoting  mills 
written  ? 

How  are  dollars  reduced  to  cents?  Dollars  to  mills? 
Cents  to  mills?  How  are  dollars  and  cents  reduced  to 
cents  ?  Dollars,  cents,  and  mills  to  mills  ?  How  are  cents 
reduced  to  dollars?  Mills  to  dollars? 

Give  the  rule  for  adding  or  subtracting  sums  of  money. 
Give  the  rule  for  multiplying  or  dividing  a  sum  of  money 
by  an  abstract  number?  By  another  sum  of  money? 

What  is  a  bill  of  goods?  What  does  it  contain? 
Against  whom  is  it  drawn?  How  is  a  bill  receipted? 
By  whom  ?  Where  are  the  dates  of  sales  written  ?  Where 
are  items  of  credit  written? 


142  INTERMEDIATE    ARITHMETIC. 

SEOTIOTST     IX. 

73f&iE&$. 


LESSON    I. 
MEASURE. 

Art.  86.  Dry  Measure  is  used  in  measuring 
grain,  fruit,  most  vegetables,  coal,  and  many  other 
dry  articles. 

The  denominations  are  pints,  quarts,  pecks, 
and  bushels. 

Table. 

2  pints  (pt.)  .  .  are  1  quart,  .  .  .  .  qt. 
8  quarts  ....  are  1  peck,  ....  pk. 
4  pecks  ....  are  1  bushel,  .  .  .  bu. 

I  bu.  =  4  pk.  =  32  qt.  *=  64  pt. 


DENOMINATE    NUMBERS.  143 

NOTES.  —  1.   The  standard  bushel  is  18}  inches  in  diameter 
and  8  inches  deep.     It  contains  2150|  cubic  inches. 

2.    In  measuring  grain,  seeds,  and  small  fruits,  the  measure 

must  be  even  full ;  but  in  measuring  corn  in  the  ear,  potatoes, 

apples,  and  other  large  articles,  the  measure  must  be  heaping 
full. 

1.  How  many  pints  in  3  quarts? 

SOLUTION.  —  In    3    quarts    there    are    3   times    2    pints, 
which  are  6  pints. 

2.  How  many  pints  in  5  quarts  ?     In  8  quarts  ? 
In  10  quarts  ? 

3.  How  many  quarts  in  10  pints  ? 

SOLUTION.  —  In   10   pints  there   are  as   many  quarts   as 

2  pints  are  contained  times  in  10  pints,  which  is  5  times. 

4.  How  many  quarts  in  8  pints  ?     In  14  pints  ? 
In  16  pints?     In  20  pints? 

5.  How  many  quarts  in  3  pecks?     In  5J  pecks? 
In  7i  pecks?     In  lOf  pecks? 

6.  How    many    pecks    in    16     quarts?     In    20 
quarts  ?     In  32  quarts  ?     In  56  quarts  ? 

7.  How    many    pecks     in    5    bushels?     In    71 
bushels?     In  9f  bushels?     In  11  bushels? 

8.  How    many    bushels    in    12    pecks?     In    20 
pecks?     In  32  pecks?     In  40  pecks? 

9.  How  many  quarts  in  8  pecks?     In  12  pecks? 

10.  How  many  pints  in  8  quarts?     In  12  quarts? 

11.  What  part   of  a  quart  is  1  pint?     2  pints? 

12.  What  part  of  a  peck  is  1  quart?     3  quarts? 

13.  What  part  of  a  bushel  is  1  peck?     2  pecks? 

3  pecks?     4  pecks?     5  pecks? 

14.  How    many    pecks    in    17    quarts?     In    27' 
quarts?     In  33  quarts? 


144        INTERMEDIATE:  ARITHMETIC. 

15.  How    many    bushels    in   13    pecks?     In    23 
pecks?     In  33  pecks? 

16.  What   will    5.V    quarts    of   plums    cost,   at   4 
cents  a  pint? 

17.  A    man    carried    3|    pecks    of    cherries    to 
market,  and  sold  them   at  10   cents   a   quart :    how 
much  did  he  receive? 

18.  If    beans    are    worth    $1.60    a    bushel,    how 
much  are  they  worth  a  quart? 

19.  When  apples  sell  at   20  cents  a  peck,  what 
are  they  worth  a  bushel  ? 

20.  A  boy  bought   half   a   bushel    of    chestnuts 
for  $1.00,  and  sold  them  at  8  cents  a  quart:   how 
much  did  he  make? 

WRITTEN    EXERCISES. 

21.  How  many  pecks  in  12  bushels?     How  many 
quarts  ?     How  many  pints  ? 

22.  Reduce  12  bu.  3  pk.  1  pt.  to  pints. 

1st  PROCESS.  2d  PROCESS. 

bu.       pk.       qt.       pt.  bu.       pk.      qt.        pt. 

12  +  3+0  +  1.  12  +  3  +  0+1. 

J  4 

48    pk.  51   pk. 

J  _-? 

51  pk.  408   qt. 

_8  _2 

408  qt.  817   pt.,    Am. 

2 

816  pt. 
_1 

817  pt.,   Ans. 


DENOMINATE    NUMBERS.  145 

23.  Reduce  5  bu.  2  pk.  7  qt.  to  pints. 

24.  Reduce  15  bu.  5  qt.  1  pt.  to  pints. 

25.  Reduce  8  bu.  3  pk.  to  quarts. 

26.  How  many  pints  in  3  pk.  5  qt.  1  pt.  ? 

27.  How  many  quarts  in  3  pk.   7  qt.  ? 

28.  How  many  pints  in  1  bu.  1  qt.  ? 

29.  How  many  bushels  in  768  pints?     817  pt.  ? 

PROCESS.  PROCESS. 

2)768  pt.  2)817  pt. 

8)384  qt.  8)408  qt.  -f  1  pt. 

4)48  pk.  4)_61  pk. 

12  bu.  12  bu.  +  3  pk. 

Am.  12  bu.  Ans.  12  bu.  3  pk.  1  pt. 

30.  Reduce  168  qt.  to  bushels. 

31.  Reduce  342  pt.  to  bushels. 

32.  Reduce  51  pt.  to  pecks. 

33.  How  many  pecks  in  37  pt.  ? 

34.  How  many  bushels  in  151  qt.  ? 

35.  What  will  3  pk.    5   qt.   of  cherries   cost,  at 
5  cents  a  pint? 

36.  A  man   sold   1   bu.    3   pk.   5    qt.   of  clover- 
seed  at  8  cents  a  quart:    what  did  he  receive? 

37.  A  fruit   dealer   paid   $7   for  3  bu.  3   pk.  of 
peaches,  and  sold  them  at  75  cents  a  peck  :   what 
was  his  gain  ? 

38.  How    many    bushels    of    chestnuts    can    be 
bought  for  $15.50,  at  5  cents  a  quart? 

39.  A    fruit    dealer   put    3   bu.    2   pk.    of  straw- 
berries   into    quart    baskets :     how    many    baskets 

were  filled  ? 
I.  A.  10. 


146  INTERMEDIATE     ARITHMETIC. 

LESSON     II. 
ZIQVI1)    MEASURE. 


Art.  87.    Liquid  Measure  is   used  in   meas- 
uring liquids ;    as?  oil,  milk,  alcohol,  etc. 

The    denominations    are    gills,    pints,    quarts, 

and    gallons. 

Table. 


4  gills    (gi) 

2   pints 

4  quarts    „     , 


are  1  pint,  . 
are  1  quart, 
are  1  gallon, 


pt. 
qt. 
gal. 


1  gal.  =  4  qt.'—  8  pt.  =  32  gi. 


NOTES.  —  1.   The  standard  liquid  gallon  contains  231  cubic 
inches. 

2.  Beer,  ale,  and   milk  were   formerly  sold   by  Beer  Meas- 
ure, the  gallon  of  which   contains  282  cubic   inches.     Beer  is 
still  sometimes  sold  by  this  measure. 

3.  The  size  of  casks  for  liquids  is  variable.     Barrels  gen- 
erally contain  31.}  gallons,  and  hogsheads  63  gallons. 


DENOMINATE    NUMBERS.  147 

1.  How  many  gills   in   3   pints?     In  10  pints? 
In  20  pints?     In  32  pints? 

2.  How  many  pints  in  5  quarts?     In  8j-  quarts? 
In  12  quarts?     In  10^   quarts? 

3.  How  many  quarts  in  5  gallons  ?     In  7f  gal- 
lons?    In  11  gallons? 

4.  How  many  pints   in  16  gills  ?     In  24  gills  ? 
In  32  gills?     In  36  gills?     In  40  gills? 

5.  How  many  quarts  in  12  pints?     In  16  pints? 
In  22  pints? 

6.  How  many  gallons  in  20  quarts?    32  quarts? 
28  quarts?     36  quarts?     40  quarts? 

7.  How   many   quarts    in    8   pints?     15    pints? 
19  pints?     13  pints?     21  pints? 

8.  How  many  pints   in   8   quarts?     11  quarts? 
16  quarts?     15^   quarts?     20  quarts? 

9.  How  many  gallons  in  8  quarts?     13  quarts? 
21  quarts  ?     24  quarts  ?     29  quarts  ? 

10.  How    many   quarts    in    6    gallons  ?     9|    gal- 
lons ?     11  gallons  ? 

11.  What    part     of    a    gallon    is    1    quart?     2 
quarts?     3  quarts?     4  quarts? 

12.  How  many  quarts  in  f  of  a  gallon? 

13.  What   will    10    quarts    of   milk    cost,   at   5* 
cents  a  pint  ? 

14.  If  a  gallon  of  wine  cost  $6,  what  will  1  pint 
cost? 

15.  If  maple    syrup    cost   $1.60    a   gallon,   what 
will  1  quart  cost  ? 

16.  At   4    cents  a  pint,   what  will  5  gallons  of 
milk  cost  ? 


148  INTERMEDIATE    ARITHMETIC. 

WRITTEN    EXERCISES. 

17.  How  many  pints  in  21  gallons  ? 

18.  How  many  gills  in  7  gal.  3  qt.  1  gi.  ? 

19.  How  many  pints  in  34  gal.  1  pt.  ? 

20.  Reduce  9  gal.  2  qt.  1  pt.  to  pints. 

21.  Reduce  38  pints  to  gallons. 

22.  Reduce  245  gills  to  gallons. 

23.  Reduce  130  gills  to  quarts. 

24.  Reduce  547  gills  to  gallons. 

25.  Reduce  45*   gallons  to  gills. 

26.  •  Reduce  56  gal.  1  pt.  to  pints. 

27.  Reduce  305  pints  to  gallons. 

28.  What  will  256  pints  of  maple  syrup  cost,  at 
$1.30  a  gallon? 

29.  How  many  vials,  holding   2   gills   each,  can 
be  filled  from  a  gallon  of  alcohol  ? 

30.  How   many  jugs,    each    containing   1   gal.  2 
qt.,  can  be  filled  from  a  barrel  of  vinegar  contain- 
ing 31^  gallons? 

31.  A  grocer  bought  25  gallons  of  maple  syrup 
at  $1.20  a  gallon,  and  sold  it  at  40  cents  a  quart : 
how  much  did  he  gain  ? 

32.  A  grocer  bought  6  barrels  of  vinegar,  con- 
taining  31*    gallons   each,   at   $6.50  a  barrel,   and 
sold   it    at   10   cents    a   quart.     Howr   much    did  he 
make  ? 

33.  A  merchant  bought  a  hogshead  of  molasses, 
containing  63  gallons,  and  sold  f  of  it  at  75  cents 
a  gallon,  and   the   rest  at  20  cents  a  quart :   what 
did  he  receive  for  it  ? 


DENOMINATE    NUMBERS.  149 


LESSON   III. 

ZOJ\TG 


Art.  88.  Long  Measitre  is  used  in  measur- 
ing lines  or  distances.  It  is  also  called  Linear 
Measure. 

The  denominations  are  inches,  feet,  yards, 
rods,  furlongs,  and  miles. 


12    inches    ( 
3    feet     . 
5J  yards 

40    rods    . 
8    furlongs 


in.) 


Table. 

are  1  foot,     . 

are  1  yard,    . 

are  1  rod,      . 

are  1  furlong, 

are  1  mile,   . 


ft. 

yd. 
rd. 
fur. 
mi. 


1  mi.  =  8  fur.  =  320  rd.  =  1760  yd.  =  5280  ft. 
63360  in. 


150 


INTERMEDIATE    ARITHMETIC. 


The  following  denominations  are  also  used  : 

e<l8llrillg 


** 


4    inches  are  1  hand,      {  ^0|^ 
3    feet       are  1  pace. 
6    feet       are  1  fathom,  { 

2    miles    are  1  league,   {  "sed    in    measuring 
^  dt  sea. 

60    geographic  miles,  or,     ) 

„„,  ..       ,         .   .    >  are  1  degree  at  the  equator. 

69J  statute  miles  (nearly),  j 

360    degrees  (  °  )  make  the  circumference  of  the  earth. 


NOTE.  —  In 
cloth,     ribbons, 
width    is    not 
and   the    yard 
into  halves,  four 
etc.      The     old 
Cloth   Measure 
used. 


measuring 

etc.,     the 

considered, 

is   divided 

h&,  eighths, 

table     of 

is   seldom 


1.  How  many  inches  in  1  foot?     In  5  feet? 

2.  How  many  feet  in  36  inches  ?     In  72  inches  ? 

3.  How  many  feet  in  4  yards  ?     In  9  yards  ? 

4.  How  many  yards  in  15  feet?     In  21  feet? 

5.  How  many  yards  in  2  rods  ?     In  6  rods  ? 

6.  How  many  yards  in  5  rods  ?     In  9  rods  ? 

7.  How  many  rods    in    2    furlongs  ?     In    5    fur- 
longs?    In  8  furlongs? 

8.  How  many  furlongs  in  80  rods?    In  120  rods? 


DENOMINATE    NUMBERS.  151 

9.    How  many  furlongs  in  6  miles?    In  9  miles? 

10.  How   many   miles    in    32    furlongs  ?     In    56 
furlongs  ?     In   72  furlongs  ? 

11.  How  many  rods  in  66  paces  ? 

12.  A   ditch    is    28    furlongs    long :    how    many 
miles  long  is  it? 

13.  A    vessel    sunk    in    water   9    fathoms   deep: 
what  was  the  depth  of  the  water  in  feet? 

14.  A   steamer   sails    3    leagues    an    hour:    how 
many  hours  will  it  take  it  to  sail  90  miles? 

15.  A    horse    is    15    hands    high:    what    is    its 
height  in  feet  ? 

16.  How  many  feet  in  a  rod? 

17.  How  many  rods  in  a  mile  ? 

18.  What  part  of  a  foot  is  9  inches? 

19.  What  part  of  a  yard  is  2  feet? 

20.  What  part  of  a  mile  is  5  furlongs  ? 

WRITTEN"    EXERCISES. 

21.  How  many  feet   in  16   yards?     How   many 
inches  ? 

22.  How  many  inches   in  3  fur.   20   rd.   3  yd.  ? 

23.  How  many  feet  in  2  fur.   30  rd.  4  ft.? 

24.  Reduce  3  mi.  5  fur.  20  rd.  to  yards. 

25.  Reduce  1650  rods  to  miles. 

26.  Reduce     32274     inches     to     higher    denom- 
inations. 

27.  Reduce  4  mi.  27  rd.  2  ft.  10  in.  to  inches. 

28.  How   many  steps  of  2  ft.   6  in.  each  will  a 
man  take  in  walking  2  miles  ? 


152 


INTERMEDIATE    ARITHMETIC. 


29.  How   many    times    will    a    wheel    6    feet    in 
circumference  turn  round  in  going  21  miles  ? 

30.  Sound    travels    1090    feet    a    second :    how 
many  miles  will  it  travel  in  60  seconds? 

31.  How  many  rods  of  fence  will  inclose  a  farm 
which  is  J  of  a  mile  long  and  |  of  a  mile  wide  ? 


LESSON    IV. 


Art.  89.  Land 
or  Square  Meas- 
ure is  used  in 
measuring  surfaces. 
It  is  also  called 
Superficial  Meas- 
ure. 

The  denomina- 
tions are  square 
inches,  square 
feet,  square  yards, 
square  rods  or 
perches,  roods,  acres,  and  square  miles. 

Art.  90.  A  Square 
Inch  is  a  square,  each 
side  of  which  is  an  inch 
in  length. 

The  figure  at  the  left  rep- 
resents a  square  inch  of  real 
size. 


b|  --". 


DENOMINATE    NUMBERS.  153 

A  Square  Yard   is   a 

square,  each  side  of  which 
is  a  yard,  or  three  feet, 
in  length.  It  contains  9 
square  feet. 

NOTE.  —  The  teacher  should 
explain  and  define  a  right  angle, 
a  square,  a  rectangle,  etc. 

Table. 

144    square  inches  (sq.  in.)  are  1  square  foot,       .     .    sq.  ff. 
9    square  feet      .     .     .     are  1  square  yard,      .     .     sq.  yd. 
30}  square  yards  .     .     .     are  1  square  rod  or  perch,  P. 

40    perches are  1  rood, E. 

4    roods are  1  acre, A. 

640    acres are  1  square  mile,      .     .     sq.  mi. 

NOTES. — 1.  Land  Surveyors  use  Gunter's  Chain,  which  is 
4  rods  or  66  feet  long,  and  consists  of  100  links,  each  link 
being  7T9o2^  inches  long.  A  square  chain  is  16  square  rods, 
and  10  square  chains  are  1  acre. 

2.  Glazing  and  stone-cutting  are  estimated  by  the  square 
foot;  painting,  plastering,  paper-hanging,  ceiling,  and  paving, 
by  the  square  yard;  and  flooring,  roofing,  tiling,  and  brick- 
laying, by  the  square  of  100  feet.  Brick-laying  is  also  esti- 
mated by  the  square  yard,  and  by  the  1000  bricks. 

1.  How  many   square   feet   in    5    square  yards? 
In  7  square  yards  ? 

2.  How  many  square  ,  yards  in   36   square  feet  ? 
In  72  square  feet?     In  90  square  feet? 

3.  How  many  perches  in  2  roods?     In  5  roods? 

4.  How    many    roods    in    80    perches  ?     In  120 
perches  ? 

5.  How  many  roods  in  8  acres?     In  12  acres? 

6.  How  many  acres  in  16  roods?     In  40  roods? 


154  INTERMEDIATE    ARITHMETIC. 

7.  How    many    square     chains     in    32    square 
rods  ?     In  64  square  rods  ?     In  80  square  rods  ? 

8.  How  many  acres  in  20  square   chains  ?     In 
40  square  chains  ?     In  80  square  chains  ? 

9.  How  many  square   yards  in   a  pavement  10 
yards  long  and  4  yards  wide? 

SOLUTION.  —  In  a  pavement  10  yards  long  and  1  yard 
wide,  there  are  10  square 'yards,  and  in  a  pavement  10 
yards  long  and  4  yards  wide,  there  are  4  times  10  square 
yards,  which  are  40  square  yards.  There  are  40  square 
yards  in  the  pavement. 

10.  How  many  square  yards  in  a  ceiling  8  yards 
long  and  6  yards  wide  ? 

11.  How  many  square   feet  in   a   board  16  feet 
long  and  lj  feet  wide? 

12.  How  many  perches  in  a  field   80   rods  long 
and  10  rods  wide  ?     How  many  roods  ? 

13.  How  many  square   inches   in   a  piece  of  tin 
15  inches  long  and  4  inches  wide  ? 

14.  How  many  square  yards  in  a  floor  15  feet 
long  and  12  feet  wide? 

WRITTEN    EXERCISES. 

15.  How    many    square    yards    in    16    perches  ? 
How  many  square  inches  ? 

16.  How  many  perches  in  5  A.  2  R.  ? 

17.  Reduce  1  A.  2  rd.  20  P.  10  sq.  yd.  7  sq.  ft. 
to  square  feet. 

18.  Reduce  70882  sq.  ft.  to  higher  denominations. 

19.  Reduce   5280   perches    to   higher   denomina- 
tions. 


DENOMINATE    NUMBERS.  155 

20.  Reduce  5184  square  inches  to  square  yards. 

21.  How  many  acres  in  a  field  56  rods  long  and 
40  rods  wide  ? 

22.  How  many  acres  in  a  street  2^  miles  long 
and  4  rods  wide  ? 

23.  How  many  square  yards  in  a  ceiling  72  feet 
long  and  40  A   feet  wide  ? 

24.  What  will   it   cost   to    pave    a  walk   60  feet 
long  and  15  feet  wide,  at  $1.25  a  square  yard? 

25.  How  many  peach  trees  can  be  planted  in  an 
orchard  containing  3  acres,  if  a  tree  be  planted  on 
each  square  rod? 

26.  If  1000  shingles  will  cover  100  square  feet, 
how  many  shingles  will  cover  a   roof  40  feet  long 
and  25  feet  wide  ? 

27.  How  many    acres    of  land  in    a  township  5 
miles  square  ? 

28.  How  many  acres  in  a  township  7  miles  long 
and  6  miles  wide? 

29.  How    many    yards     of     carpeting,     a    yard 
wide,   will   carpet   a    room   20 f    feet   long    and   18 
feet  wide  ? 

30.  How    many    bricks,    8    in.    long    and    4    in. 
wide,    will    pave    a   walk    60    feet    long    and    12 1 
feet  wide  ? 

31.  What  will    it    cost  to   plaster  the  walls   and 
ceiling   of  a  room  15  feet  long,  12  feet  wide,  and 
9  feet  high,  at  50  cents  a  square  yard? 

Art,  91.    RULE.  —  To  find  the  area  of  a  rectangle, 
Multiply  the  length  by  the  width. 


156  INTERMEDIATE    ARITHMETIC. 


LESSON  V. 
CUBIC 


Art,  92!  Cubic  Measure  is  used  in  meas- 
uring solids.  It  is  also  called  Solid  Measure. 

The  denominations  are  cubic,  inches,  cubic 
feet,  and  cubic  yards. 

A  cubic  inch  is  a  cube  whose  edges  are  each 
one  inch  long.  A  cubic  yard  is  a  cube  whose 
edges  are  each  one  yard  long. 

NOTE. — The  teacher  should  explain  and  define  a  cube; 
also  its  faces  and  edges. 

Table. 

1728  cubic  inches   (cu.  in.)   are  1   cubic  foot,    eu.  ft. 
27  cubic  feet    ....     are  1   cubic  yard,  cu.  yd. 

1  cu.  yd.  —  27  cu.  ft.  =  46656  cu.  in. 

NOTE.  —  A  cubic  yard  of  earth  is  called  a  load,  and  24 \ 
cubic  feet  of  stone  or  of  masonry  make  a  perch. 

WRITTEN    EXERCISES. 

1.  How  many  cubic  inches  in  5  cubic  feet? 
In  12  cubic  feet?  32  cubic  feet? 


DENOMINATE    NUMBERS.  157 

2.  How  many  cubic  feet  in  15552  cubic  inches? 

3.  How  many  cubic  feet  in  120  cubic  yards  ? 

4.  How  many  cubic  yards  in  405  cubic  feet  ? 

5.  Reduce  15  cu.  yd.  16  cu.  ft.  and  1305  cu.  in. 
to  inches. 

6.  Reduce  1473462  cubic  inches  to  higher  de- 
nominations. 

7.  How  many  cubic  feet  in  a  block  of  marble 
15  feet  long,  12  feet  wide,  and  5  feet  thick? 

PROCESS. 

1{-  £t  A  block  15  feet  long  and  12  feet  wide 

12  has    180    sq.   ft.    in    its    lower    face,    and 

«,          since    each    foot    of    thickness '  gives    180 
180  sq.  ft.  ,  .     f    .    „       „         c    .,  .  .          3    ...     . 

_  cubic  ieet,  five  feet  of  thickness  will  give 

5  times  180  cu.  ft.,  which  are  900  cu.  ft. 
900  cu.  ft. 

8.  How   many   cubic    feet    in    a    rock   18    feet 
long,  13  feet  wide,  and  8  feet  high  ? 

9.  How   many   cubic    feet    in    a    pile    of  wood 
24  feet  long,  3  feet  wide,  and  8  feet  high? 

10.  How   many    cubic    yards    in    a   bin    9j    feet 
long,  6  feet  wide,  and  4|  feet  deep? 

11.  How  many  cubic   feet  of  earth  must  be  re- 
moved to  make  a  cellar  44  feet  long,  27  feet  wide, 
and  5  feet  deep?     How  many  cubic  yards? 

12.  How   many    cubic    yards    of  earth    must   be 
removed  to  make  a  reservoir  120  feet  long,  54  feet 
wide,  and  9  feet  deep  below  the   surface  ? 

13.  What  will   it    cost    to    dig   a    cellar   36  feet 
long,   18f  feet  wide,  and  6i  feet  deep,  at  $2.50  a 
cubic  yard  ? 


158 


INTERMEDIATE    ARITHMETIC. 


LESSON    VI. 


Art.  93.    Wood   Measure    is    used    in    meas- 
uring  wood  and   rough   stone. 

The   denominations    are   cubic  feet,    cord   feet, 
and    cords. 


16   cubic   feet,    . 

8   cord   feet,    or, 
' 
128   cubic   feet, 


Table. 

.    are   1    cord    foot,     .     cd.  ft 

)  , 

>  are   1   cord,     .     ,     .    cd. 
I 


I  cd.  =  8  cd.  ft,  ==  128  cu.  ft. 

NOTES.  —  1.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and 
4  feet  high,  contains  1  cord  ;  and  1  foot  in  length  of  such  a 
pile  contains  one  cord  foot.  (See  cut  above.) 

2.  Formerly  40  feet  of  round  timber,  or  50  feet  of  hewn 
timber,  were  called  a  ton,  but  this  distinction  is  now  seldom 
nysofved. 


DENOMINATE    NUMBERS.  159 


WRITTEN    EXERCISES. 

1.  How   many   cord   feet  in   a  pile   of  wood  4 
feet  long,  4  feet  wide,  and  5  feet  high  ? 

2.  How  many  cubic  feet  in  6  cord  feet  ? 

3.  How  many  cord   feet  in  5J  cords   of  wood? 

4.  How  many  cords  of  wood  in  128  cord  feet? 

5.  How  many  cords  of  wood  in  a  pile  contain- 
ing 1536  cubic  feet? 

6.  How  many  cords  of  wood  in  a  pile  20  feet 
long,  4  feet  wide,  and  6  feet  high  ? 

7.  How  many  cords  of  wood  in  a  pile  48  feet 
long,  2j   feet  wide,  and  5j  feet  high? 

8.  A  man  bought  a  pile  of  wood  36  feet  long, 
4   feet   wide,   and   8    feet    high,   and    paid   $5.50   a 
cord.     What  did  the  wood  cost  him  ? 

9.  How  many  cords  of  stone  in  a  wall  40  rods 
long,  2  feet  thick,  and  4  feet  high  ? 

10.  At  $4.50  a  cord,  what  is  the  value  of  a  pile 
of  wood,  40  feet  long,  3.1,-  feet  wide,  and  6J  feet 
high? 

LESSON   VII. 

MEASURE. 


Art.  94.  Circular  Measure  is  used  in  meas- 
uring arcs  of  circles,  and  angles,  and  in  estimating 
latitude  and  longitude.  It  is  also  called  Angular 
Measure. 

The  denominations  are  seconds,  minutes,  de- 
grees, signs,  and  circumferences. 


160  INTERMEDIATE    ARITHMETIC. 

Table. 

60  seconds  ( x/ )  are  1  minute,  .  .  ' 
60  minutes  .  .  are  1  degree,  .  .  .  ° 
30  degrees  .  .  are  1  sign,  .  .  .  S. 

12  signs,  or  )  .,      .          f 

>  .     are   1   circumference,    C.  or  cir. 
360  degrees,    ) 

1  cir.  =  12  S.  =  360°  =  21600'  =  1296000" 

NOTES.  —  1.  Circular  Measure  is  used  by  surveyors  in  sur- 
veying land;  by  navigators  in  determining  latitude  and  longi- 
tude at  sea;  and  by  astronomers  in  measuring  the  motion  of 
the  heavenly  bodies,  and  in  computing  difference  in  time. 

2.  The  portion   of  surface  represented 
by  the  annexed  figure  is  a  circle. 

The  curved  line  which  bounds  the 
circle  is  its  circumference.  / 

Any  portion  of  a  circumference  is  an 
arc. 

3.  One-half  of  a  circumference  is  called 
a  semi-circumference. 

One-fourth  of  a  circumference  is  called 
a  quadrant. 

One-third  of  a  quadrant  is  called  a 
sign. 

A  semi-circumference  contains  180° ;  a 
quadrant,  90°;  and  a  sign,  30°. 

4.  Every  circumference  is  divided  into 
360  equal  parts,  called  degrees,  and,  hence, 
the  length  of  a  degree  depends  upon  the  size  of  the  circle.  A 
degree  of  the  earth's  surface  at  the  equator  contains  69£  statute 
miles,  or  60  geographical  miles  —  a  minute  of  space  being  a 
geographical  or  nautical  mile. 

1.  How  many  minutes  in   5  degrees  ? 

2.  How  many  signs  in  3  quadrants  ? 

3.  How  many  degrees  in  \  of  a  quadrant? 

4.  How  many  degrees  in  3|  signs? 

5.  How  many  signs  in  J  of  a  circumference? 


DENOMINATE    NUMBERS. 


161 


WRITTEN    EXERCISES. 

6.  How  many  seconds  in  15°  30'  ? 

7.  Reduce  15°  33'  to  minutes. 

8.  Reduce  5.1  signs  to  minutes. 

9.  Reduce  10800"  to  signs. 

10.  The  sun  appears  to  revolve  around  the  earth 
once  a  day  :  how  many  degrees  does  it  appear  to 
pass  over  in  an  hour  ?  In  6  hours  ? 


LESSON    VII. 


Art.  95.  Time 
Measure  is  used 
m  measuring  time 
or  duration. 

The  denomina- 
tions are  seconds, 
minutes,  hours, 
days,  years  and 
centuries. 


Table. 


60  seconds    (sec.) 

are  1   minute,    .     .     . 

min. 

60   minutes    .     . 

are   1   hour,         .     .     . 

h. 

24  hours   .     .     . 

are   1   day,     .... 

d. 

365  days     .     .     . 

are   1    common  year,  . 

c.  yr. 

366  days     ... 

are  1   leap  year,    .     . 

L  yr. 

100  years  (366t<L)  are  1    century, 


a 


1  d.  =  24  h.  =  1440  min.  =  86400  sec. 
I.  A.— 11. 


162        INTERMEDIATE:   ARITHMETIC. 

The  following  denominations  are  also  used : 

7   days     ....     are   1   week,   .     .     .     w. 
4  weeks        .    •.     .     are   1   lunar  month,    lr.  m. 

13  lr.  m.  1  d.  6  h.,  )  1    T  r 

0^r1     T  r   are   !   Julian  year,     /.  yr. 

or  365}   days,      j 

12   calendar  months  are   1   civil  year,    .     c.  yr. 

NOTES.  —  1.  The  exact  length  of  a  solar  year  is  365  d.  5  h. 
48  min.  48  sec.,  which  is  nearly  6  hours,  or  \  of  a  day,  longer 
than  the  common  year.  Since  the  common  year  lacks  J  of  a 
day  of  the  true  time,  an  additional  day  is  added  to  every  fourth 
year,  making  leap  year.  This  additional  day  is  given  to  Feb- 
ruary, and  hence  this  month  in  leap  year  contains  29  days. 
The  leap  years  are  exactly  divisible  by  4;  as,  1860,  1864,  1868, 
1872,  etc. 

2.  The  names  and  order  of  the  calendar  months  and  the 
number  of  days  in  each  are  as  follows: 


January,     1st  month,  31  days. 
February,  2d        "        28  or  29. 
March,       3d        "        31  days. 
April,         4th      "        30     " 
May,           5th      "        31     " 
June,          6th      "        30     " 

July,            7th  month,  31  days. 
August,        8th      "       31     " 
September,  9th      "       30     " 
October,       10th    "       31     " 
November,  llth    "       30     " 
December,   12th    "       31     " 

3.  The   following   couplet  will    assist   in    remembering  the 
months  which  have  30  days  each  : 

Thirty  days  hath  September, 
April,  June,  and  November. 

4.  In  most  business  transactions,  30  days  are  considered  a 
month,  and  360  days  a  year. 

5.  The  year  is  divided  into   four  seasons   of  three  months 
each,  as  follows  : 

March,  ATTTUMN    f  SePtember> 

SPRING,          April,  October, 

November. 


(  March, 
,          April, 
1   May. 

(  June,  C  December, 

SUMMER,  \    July,  WINTER,  I    January, 

(  August.  (.  February. 


DENOMINATE    NUMBERS.  163 

1.  How   many   seconds   in   5    minutes?     In  10 
minutes  ?     In  20  minutes  ? 

2.  How  many  minutes  in  4  hours  ?    In  8  hours  ? 

3.  How  many  hours  in  120  minutes?     In  240 
minutes  ?     In  300   minutes  ? 

4.  How  many  hours  in  3  days  ?     In  5  days  ? 

5.  How  many  days  in  48  hours?     In  72  hours? 
In  240  hours?     In  480  hours? 

6.  How  many  days  in  6  weeks  ?     In  8  weeks  ? 
In  10  weeks?     In  15  weeks? 

7.  How  many  weeks  in  35  days  ?     In  49  days  ? 

8.  How  many  weeks   in   5  lunar  months  ?     In 
12  lunar  months  ? 

9.  How  many  lunar  months   in  16  weeks  ?     In 
32  weeks  ?     44  weeks  ?     60  weeks  ? 

10.  How    many   calendar    months    in    5    years? 
In  7  years  ?     10  years  ?     12  years  ? 

WRITTEN    EXERCISES. 

11.  How  many  seconds  in  15  hours? 

12.  How  many  hours  in  28800  seconds? 

13.  Reduce  5  d.  13  h.  40  min.  to  seconds. 

14.  Reduce  31  d.  30  min.  45  sec.  to  seconds. 

15.  Reduce    30600    minutes    to    higher    denom- 
inations. 

16.  Reduce  52560  hours  to  common  years. 

17.  How  many  minutes  in  a  leap  year? 

18.  How  many  seconds  in  the  solar  year,  which 
contains  365  d.  5  h.  48.  min.  48  sec.  ? 

19.  How  many  seconds  in  a  common  year? 


164 


INTERMEDIATE    ARITHMETIC. 


20.  The   age   of  a  certain  man  is  G4  yr.   45  d. 
12  h. :    how   many    hours    has    he    lived,    allowing 
365£  days  to  the  year? 

21.  How    many    hours     in     the     three     Spring 
months  ?     In  the  three  Summer  months  ? 

22.  How    many    minutes    will    there    be    in    the 
month  of  February,  1876?     In  February,  1878? 

23.  If  your  pulse  beat  75   times  a  minute,  how 
many  times  will  it  beat  in  5  weeks? 

24.  How   many   days    will   it   take    a    steamship 
to   sail   3744    miles,   if   it   sail   at   the    rate   of  12 
miles  an  hour? 

LESSON    VIII. 
dL  r  OI21 3)  UTOIS 


1  oz.    1  Ib.    1  dr. 


Art.  96.  Avoirdupois  Weight  is  used  in 
weighing  all  articles  except  gold,  silver,  and  the 
precious  stones. 


DENOMINATE    NUMBERS.  165 

The  denominations  are  drams,  ounces,  pounds, 
hiindred-iv eights,  and  tons. 

Table. 

16  drams    (dr.)      .  are  1  ounce,  .     .     .     .  oz. 

16  ounces      ...  are  1  pound,       .     .     .  Ib. 

100  pounds     .     .     .  are  1  hundred- weight,  cwt. 

20  hundred-weights  are  1  ton,      .     .     .     .  T. 

1  T.  =  20  cwt.  =  2000  Ib.  =  32000  oz.  ==  512000  dr. 


196  pounds  of  flour,     |  ^ 

200  Ib.  pork  or  beef,    ) 

100  Ib.  of  fish are  1  quintal. 

14  Ib.   lead  or  iron are  <  1  stone. 

56  Ib.  of  corn,  rye,  or  flax-seed,  ") 

60  Ib.  of  wheat  or  clover-seed,     I  are  1  bushel. 

32  Ib.  of  oats,  J 

NOTES.  —  1.  In  wholesaling  and  freighting  coal  and  in 
invoicing  English  goods  at  the  United  States  custom-houses, 
the  hundred-weight  is  divided  into  4  quarters,  of  28  pounds 
each,  and  the  ton  contains  2240  pounds.  This  is  called  the 
long  or  gross  ton,  while  the  ton  of  2000  pounds  is  called  the 
short  or  net  ton. 

2.  The  dram  is  seldom  used  in  business  transactions,  and 
the  quarter  of  25  pounds  is  never  used. 

1.  How    many    drams    in    2    ounces?     In    5£ 
ounces?     10  ounces?     15  ounces? 

2.  How   many    ounces    in    48    drams?     In    64 
drams?     96  drams?     160  drams? 

3.  How    many    ounces    in    4    pounds?     In    6| 
pounds?     lOf  pounds?     12 J-  pounds? 

4.  How  many  pounds  in  80  ounces  ? 

5.  How    many    pounds    in    5    hundred-weight? 
In  8  cwt.?     12-|  cwt,?     25  cwt.? 


166  INTERMEDIATE     ARITHMETIC. 

6.  How   many   hundred-weight   in  4  tons?     In 
6  tons?     8§  tons?     12 '•  tons? 

7.  What    will  |  of  a    pound   of  candy  cost,  at 
2  cents  an  ounce  ? 

8.  What  will    §    of   a   hundred-weight   of  flour 
cost,  at  5  cents  a  pound? 

WRITTEN     EXERCISES. 

9.  Reduce  5  tons  to  ounces. 

10.  Reduce  3  T.  14  cwt.  56  Ib.  to  pounds. 

11.  Reduce  5  cwt.  77  Ib.  18  oz.  to  ounces. 

12.  Reduce  34920  pounds  to  tons. 

13.  Reduce    4560     ounces     to     higher    denomi- 
nations. 

14.  Reduce  11  T.  38  Ib.  15  oz.  to  drams. 

15.  What  will  a  barrel  of  flour  cost,  at  6  cents 
a  pound? 

16.  What  will  3  barrels  of  pork  cost,  at  15  cents 
a  pound? 

17.  How    many    barrels    will    3920    pounds    of 
flour  make  ? 

18.  A    farmer    sold    3600    pounds    of   wheat    at 
$1.75  a  bushel :    how  much  did  he   receive  ? 

19.  A  hay-stack  contains  9000  pounds  of  hay : 
what  is  it  worth  at  $12  a  ton? 

20.  What  will  it  cost  to  transport  50  T.  15  cwt. 
75  Ib.  of  freight,  at  \  cent  a  pound  ? 

21.  A  farmer  exchanged  45  f  pounds  of  butter, 
at   20    cents    a    pound,   for   sugar,    at   15    cents   a 
pound :   how  much  sugar  did  he  receive  ? 


DENOMINATE    NUMBERS. 


167 


LESSON   IX. 


Art.  97.  Troy 
Weiffht  is  used  in 
weighing  gold,  sil- 
ver, and  precious 
stones,  and  also  in 
philosophical  exper- 
iments. 

The  denomina- 
tions are  grains, 
pennyweights,  ounces,  and  pounds. 

Table. 

24  grains    (gr.)     .     are   1   pennyweight,     pwt. 
20  pennyweights   .     are   1   ounce,  .     .     .     oz. 
12  ounces      .     .     .     are  1   pound,       .     .     Ib. 

1  Ib.  =  12  oz.  =  240  pwt.  =  5760  gr. 

NOTES.  —  1.   Diamonds  are  weighed  by  carats  and  fractions 
of  carats,     A  carat  .is  4  Troy  grains. 


2.   The  purity  of  gold  is  also  expressed  in   carats,  a   carat 
eaning  fa  part.     Gold  that  is  22 
of  pure  gold  and  2  parts  of  alloy. 


meaning  fa  part.     Gold  that  is  22  carats  fine  contain  22  parts 
Id  ai 


1.  How  many  grains  in  5  pennyweights?     In  3 
pwt.?     8  pwt.?     101  pwt.? 

2.  Plow   many   pennyweights    in    3    ounces?     In 
6  ounces?     9  ounces?     10  ounces? 

3.  How  many  ounces  in   40   pennyweights?     In 
80  pwt,?     100  pwt?     120  pwt.? 


168 


INTERMEDIATE     ARITHMETIC. 


4.  How    many    ounces    in    4    pounds  ?     In    7| 
pounds?     12f  pounds?     20  pounds? 

5.  How  many  pounds   in    36    ounces  ?     In 
ounces  ?     84  ounces  ?     96  ounces  ? 

6.  What     part     of    a    pound    is    1 
ounces?     8  ounces?     9  ounces? 


60 


ounce  ?     6 


WRITTEN    EXERCISES. 


7.  Reduce 

8.  Reduce 

9.  Reduce 

10.  Reduce 

11.  Reduce 

12.  Reduce 

13.  Reduce 

14.  A  lady 
8  oz.  15  pwt., 
cost? 


44  Ib.  3  oz.  13  pwt.  to  pennyweights. 
7  oz.  15  pwt.  to  grains. 
56  Ib.  13  pwt.  to  grains. 
13486  pwt.  to  higher  denominations. 
40408  grains  to  higher  denominations. 
5680  ounces  to  pounds. 
5280  grains  to  ounces. 
bought  a   pearl   necklace,   weighing 
at   75    cents    a   grain  :   what   did  it 


LESSON   X. 


WEIGHT. 


Art.  98.  Apothe- 
caries Weight  is 

used  by  physicians  in 
prescribing  and  by 
apothecaries  in  mix- 
ing medicines. 

The  denominations 
are  grains,  scruples, 
drams,  ounces,  and 
pounds. 


DKNOMINATK    NUMBERS.  169 

Table. 

24  grains    (gr.}     .  are  1  scruple,      ...  9. 

3  scruples  .     .     .  are  1  dram,    ....  5. 

8  drams      ...  are  1  ounce,  ....  3. 

12  ounces     .     .     .  are  1  pound,  ....  ft). 

NOTE.  —  Medicines   are  bought   and   sold   in   quantities    by 
avoirdupois  weight. 

1.  How  many  grains  in  2  scruples  ? 

2.  How    many    scruples    in    5    drams  ?     In    7 
drams  ?     9  drams  ?     12  drams  ?     20  drams  ? 

3.  How  many    drams   in    21    scruples?     In   27 
scruples?     33  scruples?     40  scruples? 

4.  How    many    drams     in     5     ounces  ?     In     8 
ounces  ?     10  ounces  ?     12  ounces  ? 

5.  How    many  pounds   in    36    ounces  ?     In    72 
ounces?     96  ounces?     120  ounces? 

6.  How    many    ounces    in    5    pounds  ?     In    8 
pounds  ?     10-J  pounds  ?     12  pounds  ? 

WRITTEN    EXERCISES. 

7.  Reduce  16  Ib.  11 S   5  3  29   10  gr.  to  grains. 

8.  Reduce  10,1    33   to  grains. 

9.  Reduce  3563  to  pounds. 

10.  Reduce  26484  gr.  to  higher  denominations. 

11.  How  many  pounds  in  5760  3  ? 

12.  How  many  doses,  of  18  gr.  each,  in  53   29 
of  tartar  emetic? 

13.  How    many    pills,    of    5  gr.    each,    can    be 
made  from  13    23   29    of  calomel? 

14.  How  many  ounces  of  calomel  will  make  480 
pills,  each  weighing  6  grains  ? 


170 


INTERMEDIATE    ARITHMETIC. 


LESSON    XL 

S     21 


PAPER. 

24  sheets     are  1  quire. 
20  quires     are  1  ream. 

2  reams     are  1  bundle. 

5  bundles  are  1  bale. 


12  things     are  1  dozen. 
12  dozen     are  1  gross. 


12  gross       are 

20  things     are  1  score. 

NOTE.  —  A  sheet  of  paper  folded  in  2  leaves  is  called  a 
folio;  in  4  leaves,  a  quarto,  or  4to;  in  8  leaves,  an  octavo, 
or  Svo ;  in  12  leaves,  a  duodecimo,  or  12 mo;  in  18  leaves, 
an  18mo. 

1.  How  many  sheets  of  paper  in  5^   quires  ? 

2.  How  many  quires  of  paper  in  4  reams  ?     In 
8  reams?     12|  reams?     15  reams? 

3.  How    many   bundles    of   paper   in    6    reams  ? 
In  12  reams  ?     18  reams  ?     32  reams  ? 

4.  How  many  eggs  in  5  dozen  ?     In  7f  dozen  ? 
8-J  dozen  ?     12  dozen  ?     20  dozen  ? 

5.  How  many  years  are  4  score  years?     3  score 
years  and  10  ? 


WBITTEN    EXERCISES. 

6.  How  many  sheets  of  paper  in  12  i-  reams  ? 

7.  Reduce  6  rm.  15  qu.  12  sheets  to  sheets. 


DENOMINATE    NUMBERS.  171 

8.  What   will    7200    sheets    of   paper    cost,    at 
$8.50  a  ream  ? 

9.  How  many  crayons  are   there  in  36  boxes, 
if  each  box  contains  one  gross? 

10.  If  a  shirt  require  6  buttons,  how  many  shirts 
will  12  gross  of  buttons  trim  ? 

11.  What  will  44  gross    of  lead-pencils   cost,  at 
75  cents  a  dozen  ? 

12.  A  stationer  bought  15  reams  of  letter-paper 
at  $3.50  a  ream,  and   sold  it  at  25  cents  a  quire. 
How  much  did  he  gain  ? 


LESSON    XII. 
DEFINITIONS,  PRINCIPLES,  AND   KULES, 

Art.  99.  A  Denominate  Number  is  a  num- 
ber composed  of  concrete  units  of  one  or  several 
denominations. 

Art.  100.  Denominate  Numbers  are  either  Simple 
or  Compound. 

A  Simple  Denominate  Number  is  com- 
posed of  units  of  the  same  denomination ;  as, 
7  quarts. 

A   Compound   Denominate   Number  is 

composed    of  units    of   several   denominations ;    as, 
5  bu.  3  pk.  7  qt. 

NOTE.  —  Compound  Denominate  Numbers  are  properly 
called  Compound  Numbers,  since  every  compound  number  is 
necessarily  denominate. 


172  INTERMEDIATE    ARITHMETIC. 

Art.  101.  Denominate  numbers  express  Currency, 
Measure,  and  Weight. 

Currency  is  the  circulating  medium  used  in 
trade  and  commerce  as  a  representative  of  value. 

Measure  is  the  representation  of  extent,  ca- 
pacity, or  amount. 

Weight  is  a  measure  of  the  force  called  grav- 
ity, by  which  bodies  are  drawn  toward  the  earth. 

Art.  102.  The  following  diagram  represents  the 
three  general  classes  of  denominate  numbers,  their 
subdivisions,  and  the  tables  included  under  each : 

{1.  Coin,  ] 

[  United  States  Money. 
2.  Paper  Money,  J 

1.   Lines,    f  l'  LonS  Meas«re> 
or  arcs,  1  2    Circular  Measure. 


1.  Of  extension, 


2.  MEASURE,    - 


2.  Surfaces  :    Square  Measure. 


f 


1.  Cubic  Measure, 


k  3.  Capacity,      2'  Wood  Measure> 


3.  Dry  Measure, 

4.  Liquid  Measure. 


2.  Of  duration  :     Time  Measure. 
f  1.  Avoirdupois  Weight. 
3.  WEIGHT,      \  2.  Troy  Weight. 

[  3.  Apothecaries  Weight. 

Art.  103.  The  ^Reduction  of  a  denominate 
number  is  the  process  of  changing  it  from  one 
denomination  to  another  without  altering  its  value. 

Art.  104.  Reduction  is  of  two  kinds,  Reduction 
Descending  and  Reduction  Ascending. 


DENOMINATE    NUMBERS.  173 

Reduction  Descending  is  the  process  of 
changing  a  denominate  number  from  a  higher  to 
a  lower  denomination.  It  is  performed  by  mul- 
tiplication. 

'Reduction  Ascending  is  the  process  of 
changing  a  denominate  number  from  a  lower  to  a 
higher  denomination.  It  is  performed  by  division. 

Art.  105.    RULE   FOR  REDUCTION  DESCENDING. — 

1.  Multiply  the  number  of  the  highest  denomina- 
tion by  the  number  of  units  of  the  next  lower  which 
equals  a  unit  of  the  higher,  and  to  the  product  add 
the  number  of  the  lower  denomination,  if  any. 

2.  Proceed  in   like   manner  with   this   and   each 
successive   result   thus   obtained,  until   the  number  is 
reduced  to  the  required  denomination. 

NOTE.  —  The  successive  denominations  of  the  compound 
number  should  be  written  in  their  proper  order,  and  the 
vacant  denominations,  if  any,  filled  with  ciphers. 

Art.  106.    RULE    FOR   REDUCTION   ASCENDING. — 

1.  Divide    the   given    denominate    number    by   the 
number    of    units    of   its    own    denomination,  which 
equals  one  unit  of  the  next  higher,   and  place   the 
remainder,  if  any,  at  the  right. 

2.  Proceed  in    like    manner  with   this   and   each 
successive   quotient   thus   obtained,  until   the   number 
is  reduced  to  the  required  denomination. 

3.  The  last  quotient,  with  the  several  remainders 
annexed  in    proper   order,   will    be    the   answer  re- 
quired. 


174  INTERMEDIATE    ARITHMETIC. 


QUESTIONS  FOR  REVIEW. 

What  is  a  number?  What  is  an  abstract  number? 
What  is  a  denominate  number?  Into  what  two  classes 
are  denominate  numbers  divided?  Define  each. 

By  what  other  name  are  compound  denominate  num- 
bers usually  called?  Why  may  the  word  " denominate" 
be  omitted  ? 

What  is  currency?  Of  how  many  kinds  of  money  is 
United  States  currency  composed  ? 

What  is  meant  by  measure?  Name  the  two  kinds  of 
measures.  How  are  the  measures  of  extension  divided? 

What  tables  are  used  in  measuring  lines?  In  measuring 
surfaces?  In  measuring  contents?  In  measuring  capacity? 
What  table  is  used  in  measuring  duration?  What  tables 
are  used  in  measuring  the  weight  of  bodies? 

What  is  Reduction  ?  Name  the  two  kinds  of  reduction. 
Define  Reduction  Descending.  Repeat  the  rule.  Define 
Reduction  Ascending.*  Repeat  the  rule. 

For  what  is  Dry  Measure  used  ?  Name  the  denomina- 
tions. Repeat  the  table.  For  what  is  Liquid  Measure 
used?  Name  the  denominations.  Repeat  the  table.  For 
what  is  Long  Measure  used?  Name  the  denominations. 
Repeat  the  table. 

For  what  is  Square  Measure  used?  Name  the  denomi- 
nations. Repeat  the  table.  What  is  a  square  inch  ?  A 
square  yard?  How  is  the  area  of  a  rectangle  found? 

For  what  is  Cubic  Measure  used?  Name  the  denomina- 
tions. Repeat  the  table.  What  is  a  cubic  inch  ?  A  cubic 
yard?  For  what  is  Wood  Measure  used?  Name  the  de- 
nominations. Repeat  the  table. 

For  what  is  Circular  Measure  used?  Name  the  denom- 
inations. Repeat  the  table.  For  what  is  Time  Measure 
used?  Name  the  denominations.  Repeat  the  table. 

Name  the  calendar  months  in  their  order,  and  give  the 
number  of  days  in  each.  How  many  days  has  February  in 


DENOMINATE     NUMBERS.  175 

leap  years?     Name  the  four  seasons  of  the  year  and  the 
months  of  each. 

For  what  are  the  three  weights  respectively  used  ?  Give 
the  denominations  and  repeat  the  table  of  each.  Repeat 
the  miscellaneous  table. 

LESSON   XIII. 
Miscellaneous    *Review    "Problems. 

1.  How  many  quarts  in  |   of  a  bushel  ? 

2.  How  many  pints  in  3|  gallons  ? 

3.  How  many  hours  in   f   of  a  week  ? 

4.  How  many  ounces  in  2j   pounds   of  sugar? 
In  2i  pounds  of  silver? 

5.  What  will   §   of  a   cwt.  of  sugar  cost,  at  15 
cents  a  pound  ? 

6.  What  will  |  of  a   gallon   of  oil   cost,  at  25 
cents  a  pint  ? 

7.  What  will  |  of  a  ream  of  paper  cost,  at  20 
cents  a  quire  ? 

8.  What  will  §  of  a  ton  of  hay  cost,  at  75  cents 
a  cwt.  ? 

9.  A  boy  picked  3  pecks  of  cherries,  and  sold 
them  at  10  cents  a  pint :  how  much  did  he  receive  ? 

10.  If  a    ship   sail    3    leagues    an   hour,   in   how 
many  hours  will  it  sail   63  miles? 

11.  How  many  half-pint  bottles  will  a  gallon  of 
sweet  oil  fill  ? 

12.  How  many  quart  baskets  will  3  pk.  5  qt.  of 
strawberries  fill  ? 

13.  How  many  leap  years  in  every  century  ? 

14.  How  many  calendar  months  in  20  years  ? 


176  INTERMEDIATE    ARITHMETIC. 


WRITTEN    EXERCISES. 

15.  A  fruit  dealer  bought  24  barrels   of  apples, 
containing  2f  bushels  each,  at  $2.50  a  barrel,  and 
sold  them  at  $1.25  a  bushel :    what  was  his  gain? 

16.  What  will  20  yd.  2  ft.  of  iron  railing  cost,  at 
$1.25  a  foot? 

17.  What  will  40  miles  of  telegraph  wire  cost,  at 
25  cents  a  yard  ? 

18.  How  many   times   will   a    carriage- wheel    11 
feet    in    circumference    turn    round    in    running    2 
miles  ? 

19.  How  many  times  will  a  car-wheel    5  feet  in 
circumference    turn  round  in  running  from  Colum- 
bus to  Cincinnati,  the  distance  being  120  miles? 

20.  How    many    acres    in    a    township    6    miles 
square  ? 

21.  What  will  a  piece  of  land  40  rods  long  and 
32  rods  wide  cost,  at  $75  an  acre  ? 

22.  How  many  hills    of  corn  can  be  planted  on 
5  acres,  allowing  1  hill  to  every  square  yard? 

23.  How  many  people    can    stand    on   a   terrace 
250  feet  long  and  120  feet  wide,  allowing  4  persons 
to  each  square  yard? 

24.  What  will  it  cost  to  gravel  a  street  129  rods 
long  and  60  feet  wide,  at  75  cents  a  square  yard  ? 

25.  How  many  square  yards  of  plastering  in  the 
walls  and   ceiling  of  a  room  21  feet  long,  18  feet 
wide,  and  9  feet  high  ? 

26.  How  many  yards  of  carpeting,  a  yard  wide, 
will  carpet  a  room  18  *  feet  long  and  15  feet  wide  ? 


DENOMINATE    NUMBERS.  177 

27.  If  1000  shingles  will  cover  100  square  feet, 
how  many  shingles  will  cover  a   roof  each  side  of 
which  is  48  feet  long  and  15  feet  wide? 

28.  A  park  containing  40  acres  is  50  rods  wide : 
how  long  is  it  ? 

29.  How  many  cubic  feet  in  a  bin  12  feet  long, 
8  feet  wide,  and  3|  feet  deep  ? 

30.  How   many    perches    of   stone   in  a  wall   99 
feet  long,  8  feet  high,  and  1J-  feet  thick? 

31.  What   will   it   cost  to    dig   a   ditch   80   rods 
long,  4\  feet  wide,  and  2  feet  deep,  at  15  cents  a 
cubic  yard? 

32.  At  $4.50  a  cord,  what  will  be  the  cost  of  a 
pile  of  wood  48  feet  long,  6  feet  high,  and  4  feet 
wide  ? 

33.  How   many    times    will    a    clock    that    ticks 
seconds,  tick  in  the  month  of  June  ? 

34.  If  a  person  read  a  half  hour  each  day,  how 
many   hours    will   he    read    in    40    years,  of    365 \ 
days  each  ? 

35.  How    many    gold    rings,    each    weighing    4 
pwt.,    can   be    made    from    a    bar  of  gold  weighing 
1  Ib.  4  oz.  ? 

36.  A  car  contains  80  barrels  of  pork,  and  an- 
other 80  barrels  of  flour :  what  is  the  diiference  in 
the  freight  of  the  two  cars  ? 

37.  How  many  gross    of  pens  will   supply  4320 
pupils  one  year,  if  each  pupil  require  4  pens? 

38.  If  10    sheets    of   paper   will    make    a   16mo 
book  of  320  pages,  how  many  reams  will  it  take  to 
publish  an  edition  of  2000  copies  ? 

I.  A.— 12. 


178  INTERMEDIATE    ARITHMETIC. 


SECTION     X. 


LESSON   I. 
Addition    of  Compound  Numbers. 

1.  What  is  the  sum  of  5  bu.  3  pk.  6  qt.  1  pt.  ; 
8  bu.  2  pk.  1  pt.  ;  10  bu.  1  pk.  3  qt.  ;  and  3  pk. 
5  qt.  1  pt.  ? 

PROCESS.  Write  the   compound  num- 

bu.       pk.       qt.       pt.  bers  so  that  terms  of  the  same 

5361  denomination    shall     stand    in 

8201  the    same   column.     Add   first 

10  3  the  column  of  pints.     The  sum 

_  _  3          5          1  js   3  pints,  which    equal    1  qt. 

25  bu,  2  pk.  7  qt,    1  pt.  1  pt.     Write    the   1    pt.    under 

the    pints,   and    add   the   1  qt. 

with  the  column  of  quarts.  The  sum  of  the  quarts  is  15 
quarts,  which  equal  1  pk.  7  qt.  Write  the  7  qt.  under  the 
quarts,  and  add  the  1  pk.  with  the  column  of  pecks.  The 
sum  of  the  pecks  is  10  pecks,  which  equal  2  bu.  2  pk. 
Write  the  2  pk.  under  the  pecks,  and  add  the  2  bu.  with 
the  column  of  bushels.  The  sum  of  the  bushels  is  25 
bushels.  The  sum  of  the  four  compound  numbers  added 
is  25  bu.  2  pk.  7  qt.  1  pt, 

(2)  (3)  (4) 

bu.  pk.  qt.   pt.  gal.  qt.    pt.   gi.  mi.   fur.  rd.  yd.   ft.    in. 

16     2     6      1  21      3      1      3  19     7  39  5      2     10 

23      1      4     0  16     0     1      2  27     3  24  3      1       6 

40     3     0      1  48     2     0     0  45     4  33  0     0      7 

9020  35     013  60  17  210 


COMPOUND    NUMBERS.  179 


(5) 

cwt.  Ib.  oz.  dr. 
15  63  11  13 
18  85   0  10 
6  15  15   0 

(6) 
Ib.  oz.  pwt.  gr. 
9  11  19  23 
13   6  13  20 
7  10   8  11 

(7) 
».  I-  3-  9-  gr, 
44  11  7  2  19 
23   9  6  0   8 
10  5  2  16 

0     75      87                    9     15     16  27       7     6     1     14 

19    36     14     15          23      0     10 9  16      300     18 

8.  What  is  the  sum  of  15  w.  5  d.  22  h.  45  min. 
34  sec. ;    8  w.    6  d.   13  h. ;    3  w.    20  h.    52  min. ; 
4  d.  22  h.  33  min.  55  sec.;    1  w.  2  d.  3  h.  30  min.? 

9.  Add  14°  30'  46";   53°  16'  49";   26°  34'  15"; 
18°  44'  33";   62°  36';    and  43°  45". 

10.  Add  5  sq.  mi.  625  A.  3  R.  35  P. ;   14  sq.  mi. 
546  A.   2  R.   28  P.;     486  A.   1  R.   27  P. ;    94  A. 
24  P. ;    and  14  sq.  mi.  300  A.  3  R.  36  P. 

11.  A  wood  dealer  bought  5  piles   of  wood,  the 
first  containing  21  cd.  5  cd.  ft.  15  cu.  ft. ;    the  sec- 
ond, 45  cd.  12  cu.  ft.;    the  third,  18  cd.   7  cd.  ft.; 
the    fourth,    50  cd.    6  cd.  ft.    14  cu.  ft. ;    and    the 
fifth,    16  cd.    5  cd.  ft.      How    much    wood    did   he 
purchase  ? 

12.  A  printer  used  3  bundles   1  ream   16  quires 
of  paper  on  Monday ;    2  bundles  1  ream  on  Tues- 
day;    4  bundles   16  quires    on  Wednesday;    3  bun- 
dles 1  ream  18  quires  on  Thursday ;    5  bundles  on 
Friday ;    and  3  bundles  1  ream  on  Saturday.     How 
much  paper  did  he  use  during  the  week? 

13.  The  four  quarters  of  an  ox  weighed  respect- 
ively 2  cwt.   84  Ib.   10  oz.;    3  cwt.    1  Ib.   14  oz. ; 
2  cwt.    76  Ib.    4  oz.;     and    2  cwt.    98  Ib.    14  oz. 
What  was  the  weight  of  the  four  quarters? 


180  INTERMEDIATE    ARITHMETIC. 

14.  A  garden  has  four  unequal  sides.     The  first 
is  4  rd.   3  yd.   2  ft.   8  in. ;    the   second,   5  rd.  1  ft. 
10  in. ;     the    third,    4  rd.    5  yd.    4  in. ;     and    the 
fourth,   3  rd.   4  yd.   2  ft.    9  in.     What   is   the    dis- 
tance round  the  garden  ? 

15.  A   cistern   full   of  water  was   emptied   by  3 
pipes.     The  first  discharged  45  gal.  3  qt. ;    the  sec- 
ond, 54  gal.  1  pt. ;    and  the  third,  61  gal.  2  qt.  1  pt. 
How  much  water  did  the  cistern  contain  ? 

DEFINITIONS,  PKINCIPLE,  AND  KULE, 

Art.  107.  A  Compound  Number  is  a  num- 
ber composed  of  units  of  several  denominations. 

Art.  108.  The  numbers  expressing  the  succes- 
sive denominations  of  a  compound  number,  are 
called  its  Terms. 

Compound  numbers  are  of  the  same  kind  when 
their  corresponding  terms  express  units  of  the  same 
denomination  ;  as,  3  bu.  2  pk.,  and  6  bu.  3  pk.  5  qt. 

Art.  109.  Compound  Addition  is  the  pro- 
cess of  finding  the  sum  of  two  or  more  compound 
numbers  of  the  same  kind. 

Art.  110.  PRINCIPLE.  —  In  both  simple  and  com- 
pound addition,  the  sum  of  each  column  is  divided 
by  the  number  of  units  of  that  denomination,  which 
equals  one  of  the  next  higher  denomination.  In 
simple  addition  this  divisor  is  10;  in  compound 
addition  it  is  a  varying  number,  since  the  several 
denominations  are  expressed  on  a  varying  scale. 


COMPOUND    NUMBERS.  181 

Art.  111.  RULE.  —  1.  Write  the  compound  num- 
bers to  be  added  so  that  units  of  the  same  denomina- 
tion shall  stand  in  the  same  column. 

2.  Add  first  the  column  of  the  lowest  denomina- 
tion^ and  divide  the  sum  by  the  number  of  units  of 
ihat  denomination,  ivhich  equals  a  unit  of  the  next 
higher    denomination;    write    the    remainder    under 
the    column    added,   and    add   the   quotient   with   the 
next  column. 

3.  In   like  manner  add    the   remaining  columns, 
writing  the  sum  of  the  highest  column  under  it. 

LESSON    II. 
Subtraction  of  Compound  Numbers. 

1.   From   13  Ib.   5  oz.   16  pwt.  21  gr.   take   9  Ib. 

4  oz.   18  pwt.   15  gr. 

PROCESS.  Write  the  subtrahend  under 

Ib.  oz.  pwt.  gr.  the  minuend,  placing  terms  of 

13  5  16  21  the  same  denomination  in  the 

9        4        18        15  same   column.     Subtract  15  gr. 

4  Ib.  0  oz.  18  pwt.  6  gr.  from  21  gr.,  and  write  6  gr., 
the  difference,  under  the  grains. 

Since  18  pwt.  are  greater  than  16  pwt.,  add  20  pwt.  to 
16  pwt.,  making  36  pwt.  Subtract  18  pwt.  from  36  pwt., 
and  write  18  pwt.,  the  difference,  under  the  pennyweights. 
Since  20  pwt.  were  added  to  the  minuend,  add  1  oz.  (which 
equals  20  pwt.)  to  the  4  oz.  of  the  subtrahend,  making 

5  oz.     Subtract  5  oz.  from  5  oz.,  and  write  0  oz.,  the  dif- 
ference, under  the  ounces.     Subtract  9  Ib.  from  13  Ib.,  and 
write   4  Ib.,  the  difference,  under  pounds.     The  difference 
is  4  Ib.  18  pwt.  6  gr. 


182  INTERMEDIATE    ARITHMETIC. 


(2) 

(3) 

(4) 

cwt. 

Ib. 

oz. 

dr. 

Ib. 

g. 

3- 

9- 

gr. 

w. 

d. 

h.  min. 

From  48 

73 

10 

15 

7 

10 

1 

i 

14 

13 

1 

13  45 

Take  29 

47 

14 

9 

3 

11 

5 

2 

16 

8 

6 

17  33 

(5)  (6)  (7) 

mi.    fur.    rd.  yd.  rd.  yd.    ft*  in.  gal.  qt.   pt.  gi. 

From  405     5     25  4  35  5     2  10  44  3     1     2 

Tto    384     6     37  5  27  4      1  11  26  3     1     3 


8.  A  farmer  raised  7  bu.  1  pk.  4  qt.  of  clover- 
seed,  and   sold   5  bu.   6  qt.   1  pt.     How  much   had 
he  left? 

9.  A   man   bought    a    farm    containing    356  A. 

2  R.    25  P.,    and    sold    148  A.    3  R.    36  P.     How 
much  land  had  he  left? 

10.  Washington  is  77°  2'  48"  W.  longitude,  and 
San   Francisco    122°  26'  15"   W.   longitude.      How 
much    farther   west   is    San   Francisco   than  Wash- 
ington ? 

11.  From  a  stack  of  hay,  containing  5j  tons,  a 
farmer  sold  3  T.   12  cwt.   65  Ib.      How  much   hay 
remained  unsold  ? 

12.  From    a    hogshead    of    molasses,    containing 
63  gallons,  a  grocer  sold  38  gal.  3  qt.  1  pt.      How 
much  molasses  remained  in  the  hogshead? 

13.  A  silversmith  bought  a  bar  of  gold  weighing 
1  Ib.    5  oz.    12  pwt.,   and   a   bar  of  silver  weighing 

3  Ib.  8  oz.  16  pwt.  10  gr.      How  much  more  silver 
than  gold  did  he  buy  ? 

14.  A     company     contracted    to     build     65  mi. 

4  fur.   of   railroad,   and    completed    the    first    year 


COMPOUND    NUMBERS.  183 

27  mi.   7  fur.   20  rd.      How   much    remained  to   be 
"built? 

15.    A  note  was  given  July  23,  1863,  and  paid 
Sept.  15,  1869.      How  long  did  it  run? 

PROCESS.  Write  the  earlier  date  under  the 

,  later,   writing    the    number   of  'the 

year,    month,    and    day    in    proper 
order,    and    subtract,     allowing    30 

7  9Q 

'         ^  days    to    a   month    and   12   months 


6  yr.    1  mo.  22  d.        to   a  year. 

16.  What    is    the    difference    of    time    between 
Oct.   23,   1856,   and  June   15,   1866? 

17.  How    long    from    April    12,    1861,    to    May 
22,   1865. 

18.  Abraham    Lincoln  was  born  Feb.  12,  1809, 
and   died  April  15,  1865.     What  was  his   age? 

19.  The    American    Revolution   began    April  19, 
1775,  and  ended  Jan.  20,  1783.     How  long  did  it 
continue  ? 

20.  America  was  discovered  Oct.  14,  1492,  and 
the  Declaration  of  Independence  was   signed  July 
4,  1776.     How  much   time    elapsed  between  these 
two  events  ? 

21.  The  laying   of  the  Atlantic   Cable  was  con- 
summated July  28,  1866,  and  the  Pacific  Railroad 
was   completed  May  10,  1869.     How  much  earlier 
was  the  first  event  than  the  second? 

22.  Andrew   Jackson    died    at   Nashville,  Tenn., 
June   8,  1845,   aged   78  yr.  2  mo.  23  days.     What 
was  the  date  of  his  birth? 


184  INTERMEDIATE    ARITHMETIC. 

DEFINITION  AND  EULE, 

Art.  112.    Compoiind    Subtraction    is    the 

process  of  finding  the  difference  between  two  com- 
pound numbers  of  the  same  kind. 

Art.  113.  RULE.  —  1.  Write  the  subtrahend  under 
the  minuend,  placing  terms  of  the  same  denomina- 
tion in  the  same  column. 

2.  Beginning  at  the  right,  subtract  each  successive 
term  of  the  subtrahend  from  the  corresponding  term 
of  the  minuend,  and  write  the  difference  beneath. 

3.  If   any    term    of    the    subtrahend    be    greater 
than   the   corresponding   term   of  the  minuend,   add 
to  the  term   of  the   minuend  as  many  units  of  that 
denomination  as  equal  one  of  the  next  higher,  and 
from  the  sum  subtract  the  term  of  the  subtrahend, 
writing  the  difference  beneath. 

4.  Add  one  to  the  next  term   of  the  subtrahend, 
and  proceed  as  before.  < 

NOTE.  —  Instead  of  adding  one  to  the  next  term  of  the 
subtrahend,  one  may  be  subtracted  from  the  next  term  of 
the  minuend. 

LESSON   III. 

Multiplication  of  Compound  Numbers. 
1.    Multiply  34  gal.   3  qt.   1  pt.  by  9. 

PROCESS  Write  the   multiplier  under  the 

lowest   denomination    of   the    mul- 

34  gal.  3  qt.   1  pt.         tiplicand.     9  times  1  pt.  are  9  pt., 
equal  to  4  qt.  1  pt.     Write  the  1  pt. 


313  gal.  3  qt.   1  pt.         under  pints,  and  reserve  the  4  qt. 


COMPOUND    NUMBKRS.  185 

to  add  to  the  product  of  quarts.  9  times  3  qt.  are  27  qt., 
and  4  qt.  added  are  31  qt.,  equal  to  7  gal.  3  qt.  Write 
the  3  qt.  under  quarts,  and  reserve  the  7  gal.  to  add  to 
the  product  of  gallons.  9  times  34  gal.  are  306  gal.,  and 

7  gal.  added  are  313  gal.     Hence,  9  times  34  gal.  3  qt.  1  pt. 
—  313  gal.  3  qt.   1  pt. 

(2)                                      (3)                                (4) 
15  Ib.  6  oz.  13  pwt.        7  yd.  2  ft.  11  in.        15  mi.  -3  fur.  22  rd. 
8  12  6 

(5)  (6)  (7) 

8  ft).  10  g    4  Z    2  9.        27  bu.  3  pk.  5  qt.        4  w.  6  d.  13  h. 
4  9  7_ 

8.  If   a   barrel  of   sugar   weighs    2  ewt.   45  Ib. 
8  oz.,  how  much  will  6  barrels  weigh? 

9.  How  much  gold   will  make    a   dozen   rings, 
each  weighing  7  pwt.  15  gr.  ? 

10.  If   a   pupil    studies   4  h.    30  min.    each   day, 
how  many  hours  will  he  study  in  12  school  weeks, 
of  5  days  each  ? 

11.  If  a  ship  sail  3°  25'  33"  in   one   day,  how 
far  will  it  sail  in  15  days  ? 

12.  If  one   man   can   build   5  rd.  4  yd.    2  ft.    of 
fence  in  a  day,  how  many  yards  can  8  men  build? 

13.  How   much  wheat   in   12    bins,   if   each   bin 
contains  50  bu.  2  pk.  5  qt.  ? 

14.  John's    age   is    7  yr.    9  mo.   16  d.,   which   is 
one-fifth  of  the  age  of  his  father  :    how  old  is  his 
father? 

15.  If  a  load  of  wood  contains  6  cd.  ft.  12  cu.  ft., 
how  much  wood  will  15  loads  make? 


186  INTERMEDIATE    ARITHMETIC. 

16.  If  a   family   use    2   gal.    3  qt.   1  pt.   of  milk 
a  week,  how  much  will  it  use  in  a  year? 

17.  How  much    hay.  is   there   in    6   stacks,   each 
containing  4  T.  16  cwt.   70  Ib.  ? 

18.  What   is   the   distance  round  a  square   field, 
each  side  of  which  is  24  rd.  3  yd.  2  ft.  ? 

19.  If  a  printer  uses  3  reams  15  quires  12  sheets 
of  paper  each  day,  how  much  paper  will  he  use  in 
4  weeks,  of  6  days  each  ? 

20.  If  a  man  walk  2  mi.  7  far.  32  rd.  an  hour, 
how  far  will  he  walk  in  12  hours  ? 

21.  A  field  contains  25   rows  of  corn.     If  each 
row    yield    5  bu.    3  pk.,    how   much    corn    will    the 
field  yield? 

DEFINITION  AND  KULE, 

Art.  114.    Compound  Multiplication  is  the 

process    of    taking    a    compound    number    a    given 
number  of  times. 

Art.  115.  RULE.  —  1.  Write  the  multiplier  under 
the  lowest  denomination  of  the  multiplicand. 

2.  Beginning  at  the  right,  multiply  each  term  of 
the  multiplicand  in  order,  and  reduce  each  product 
to  the  next  higher  denomination,  writing  the  re- 
mainder under  the  term  multiplied,  and  adding  the 
quotient  to  the  next  product. 

NOTE — In  both  simple  and  compound  multiplication  the  suc- 
cessive products  are  each  divided  by  the  number  of  units  of  their 
denomination,  which  equal*  one  of  the  nex'  higher  denomination. 


COMPOUND    NUMBERS.  187 

LESSON   IV. 
division  of  Compound  Numbers. 

1.   Divide  15  w.  6  d.  13  h.  12  min.  by  12. 

PROCESS.  Write  the  divisor  at  the 

0  ,     ., »      .  left    of   the    dividend,    as 

12)15  w.  6  d.  13  h.  12  mm.          .       .       .      ,.   .  . 

in  simple  division,     -fa  or 
1  w.  2  d.     7  h.     6  min.         15  w  =  l  w    with  3  w>  re_ 

maining.     Write  the    1  w. 

under  weeks.  The  3w.  remaining  equal  21  d.,  and  21  d. 
and  6  d.  equal  27  d.  TV  of  27  d.  =  2  d.,  with  3  d.  remain- 
ing. Write  the  2  d.  under  days.  The  3  d.  remaining 
equal  72  h.,  and  72  h.  and  13  h.  equal  85  h.  TV  of  85  h. 
=  7  h.,  with  1  h.  remaining.  Write  the  7  h.  under  hours. 
The  1  h.  remaining  equals  60  min.,  and  60  min.  and 
12  min.  equal  72  min.  T^  of  72  min.  —  6  min.  Write 
the  6  min.  under  minutes.  The  quotient  is  1  w.  2  d. 
7  h.  6  min. 

(2)  (3)  (4) 

8)141b.  12  oz.  15  dr.    10)  53  yd.  2ft.  8  in.    5)52A.  3R.  30P. 

(5)  (6)  (7) 

6 )  9cwt.  731b.  12oz.      11 )  65w.  Id.  Ih.  58min.     7 )  14flx  5§  65. 

8.  If  a   man  sleep    52  h.    30  min.    in   a   week, 
how    long    does    he    sleep,    on    an    average,    each 
day? 

9.  A  man   bought   a   stack   of  hay,   containing 
6  T.  19  cwt.  86  lb.,   and   drew  it  home  in  7  equal 
loads.    How  much  hay  did  he  draw  at  each  load? 

10.    If  a  dozen  silver  spoons  weigh  8  oz.  15  pwt.r 
what  is  the  weight  of  each  spoon  ? 


188  INTERMEDIATE    ARITHMETIC. 

11.  A  farm   of  345  A.  3  R.   24  P.  was   divided 
equally  between  6   heirs  :   ho\v  much   did   each  re- 
ceive ? 

12.  Five    equal    casks    of   vinegar    contain    218 
gal.   2  qt.  :    how  much  vinegar  in  each  cask  ? 

13.  If  a  man  can  dig  a  ditch  36  rd.  4  yd.  2  ft. 
long  in  8  days,  how  much  can  he  dig  in  1  day  ? 

14.  If  9  men  can  pave   22  sq.  rd.   25  sq.  yd.  in 
a  day,  how  much  can  1  man  pave  in  a  day  ? 

15.  A  ship  sailed  48°  24r  45"  in  15  days  :   how 
far  did  it  sail  each  day  ? 

16.  How   many   goblets    can   be    made    of   5  Ib. 
6  oz.  12  pwt.  of  silver,  if  each  goblet  weighs  7  oz. 
8  pwt.  ? 

PROCESS.  Keduce    both    divi- 

5  Ib.  6  oz.  12  pwt  =  1332  pwt.          dend    and    divisor   to 

pennyweights,  and  di- 
7oz.     8pwt.=     148  pwt,  y' 


1332  pwt,  -^-  148  pwt,  =  9,  Am.          vision. 

17.  How  many  bottles,  holding  3  qt.  1  pt.  each, 
can  be  filled  from  a  cask  containing  45  \  gallons? 

18.  How  many   baskets    of   peaches,    containing 

3  pk.  4  qt.  each,  will  make  3J-  bushels? 

19.  How  many  lengths  of  fence,  each  10  ft.  4  in., 
will  make  28  rd.  3  ft.  of  fence  ? 

20.  How   many    castings,    weighing   12  Ib.    8  oz. 
each,  can  be  made  from  5  cwt.  50  Ib.  of  iron? 

21.  How  many  times  will  a  wagon  wheel,  11  ft. 
8  in.  in   circumference,  turn  round  in   going  2  mi. 

4  fur.  ? 


COMPOUND    NUMBERS.  189 

22.  How    many    rings,   weighing    5  pwt.    16  gr. 
each,   can   be   made   from   a  bar   of  gold  weighing 
1  Ib.  8  oz.? 

23.  How  many  steps,  of  2  ft.  4  in.  each,  will  a 
man  take  in  walking  }  of  a  mile? 

24.  If  a   man  can  walk  2  m.  6  fur.  in  an  hour, 
how  long  will  it  take  him  to  walk  22  miles? 

DEFINITION  AND   KULES. 

Art.  116.  Compound  Division  is  the  pro- 
cess of  dividing  a  compound  number  into  equal 
parts. 

Art.  117.  RULE  I.  —  1.  Write  the  divisor  at  the 
left  of  the  dividend,  as  in  simple  division. 

2.  Beginning  at  the  left,  divide  each  term  of  the 
dividend    in    order,    and    ivrite    the   quotient   under 
the  term  divided. 

3.  If  the  division  of  any  term  give  a  remainder, 
reduce  it  to  the  next  lower  denomination,  to  the  re- 
sult  add   the  number  of  that   denomination   in   the 
dividend,  and  then  divide  as  above. 

NOTE.  —  When  the  divisor  is  a  large  number,  the  succes- 
sive terms  of  the  quotient  may  be  written  at  the  right  of  the 
dividend,  as  in  long  division. 

Art.  118.  RULE  II. —  To  divide  a  compound  num- 
ber by  another  of  the  same  kind,  Reduce  both  com- 
pound numbers  to  the  same  denomination,  and  then 
divide  as  in  simple  division. 

NOTE. — This  is  not  properly  compound  division,  since  the 
compound  numbers  are  reduced  to  simple  numbers  before 
dividing. 


190  INTERMEDIATE    ARITHMETIC. 

v 

LESSON   V. 
Miscellaneous   "Problems. 

1.  If  5  sheets  of  copper  contain  28  Ib.  10  oz. 

8  dr.,  how  much  copper  is  there  in  each  sheet? 

2.  How  much  silver  will  it  take  to  make  4  dozen 
spoons,  each  spoon  weighing  15  pwt.  12  gr.  ? 

3.  If  a  milk  dealer  sells  daily  7  cans  of  milk, 
each  holding  12  gal.  2  qt.,  how  much  milk  does  he 
sell  in  4  weeks  ? 

4.  From    the   sum    of  15  Ib.  8  oz.  15  pwt.  and 

9  Ib.  10  oz.  18  pwt.  take  their  difference. 

5.  John    Jones    was    born    Aug.   8,   1856,    and 
on  Jan.  1,  1862,  his   age  was  just  \  of  the  age  of 
his  father.     How  old  was  his  father? 

6.  Two  small  casks,  each  holding  21  gal.  3  qt., 
were  filled  from  a  cask  of  cider  containing  56  gal. 
2  qt.    How  much  cider  remained  in  the  large  cask? 

7.  A  father  owning  a  farm  of  256  A.  3  R,  24  P., 
gave  100  A.  to  his  son  and  then  divided  what  re- 
mained equally  between  his  two  daughters.     What 
was  each  daughter's  share  ? 

8.  A  farmer  having   cut   12  T.    15  cwt.  of  hay 
from  a  meadow,  sold  6  loads  of  1  T.  3  cwt.  75  Ib. 
each,  and  then  put  the  rest  in  a  stack.     How  much 
hay  was  in  the  stack  ? 

9.  A    merchant   bought    3    chests    of  tea,    each 
weighing    2  cwt.    45  Ib.,    and    in    one    month   sold 
4  cwt.  80  Ib.  12  oz.     How  much  tea  had  he  left? 

10.   A   publisher   bought    20    bundles    of   paper, 


COMPOUND    NUMBERS.  191 

and  used   daily  3  reams  15  quires    12  sheets  :    how 
much  paper  had  he  left  at  the  close  of  12  days  ? 

11.  A   railroad    company    bought    145    cords    of 
wood,   piled   in   three    ranks ;    the   first    rank    con- 
tained   36  cd.    5  cd.  ft. ;     and    the    second,    64  cd. 
6  cd.  ft.    12  cu.  ft.     How   much   wood   was    in    the 
third  rank  ? 

12.  A  man  bought  3  loads   of  hay,  which,  with 
the  wagon,  weighed  respectively  1  T.  8  cwt.  40  lb.; 

I  T.  11  cwt.   80  lb. ;    arid   1  T.   9  cwt.  60  lb. ;    and 
the  wagon  alone  weighed  10  cwt.  90  lb.     How  much 
hay  did  he  buy? 

13.  If  4   horses   eat   15  bu.    3  pk.   4  qt.  of  oats 
in  12  days,  how  much  will  they  eat  in  one  day? 

14.  If  5  horses    eat   21  bu.   1  pk.    6  qt.    of  oats 
in   4   weeks,    how  much    will    3    horses    eat   in   the 
same  time  ? 

15.  How  many  steps,  of  2  ft.   6  in.  each,  will  a 
man  take  in  walking  4  fur.  20  rd.  ? 

16.  How    many    times    will    a     carriage    wheel, 

II  ft.   4  in.   in    circumference,   turn   round  in  run- 
ning 10  miles? 

17.  How  many  yards  of  carpeting,  a  yard  wide, 
will  carpet  a  room  18  ft.  by  21  ft.  ? 

18.  How    many    yards     of    Brussels    carpeting, 
*   of  a   yard   wide,   will    carpet    a   room    20  ft.    by 
28  fr.  ? 

19.  Three  men,  A,  B,  and  C,  bought  a  hogshead 
of  sugar,  weighing  13  cwt.  60  lb. ;   A  received  -J-  of 
it,    B   §    of   the    remainder,   and   C    what  was   left. 
How  much  sugar  did  each   receive  ? 


192  INTERMEDIATE    ARITHMETIC. 

20.  A  company  graded  25  mi.  5  fur.  36  rd.  of 
road;  *  of  the  job  was  completed  the  first  month, 
\  of  it  the  second  month,  |  of  it  the  third  month, 
and  the  rest  the  fourth  month.  How  many  miles 
of  road  were  graded  each  month  ? 


QUESTIONS  FOR  REVIEW. 

What  is  a  simple  number?  What  is  a  compound  num- 
ber? When  is  a  simple  number  denominate?  When  con- 
crete? When  abstract? 

What  is  meant  by  the  terms  of  a  compound  number? 
When  are  two  or  more  compound  numbers  of  the  same 
kind?  Give  examples. 

What  is  compound  addition  ?  In  what  respect  does  it 
differ  from  simple  addition?  Give  the  rule. 

What  is  compound  subtraction?    Repeat  the  rule. 

How  do  you  find  the  difference  of  time  between  two 
dates?  How  many  days  to  the  month  are  allowed? 

What  is  compound  multiplication?  How  does  it  differ 
from  simple  multiplication?  Repeat  the  rule. 

What  is  compound  division?  How  does  it  differ  from 
simple  division?  Repeat  the  rule? 

How  is  one  compound  number  divided  by  another? 
Why  must  the  two  compound  numbers  be  of  the  same 
kind?  Is  the  process  simple  or  compound  division? 


UNIVERSITY   OF    CALIFORNIA 
LIBRARY 

is  is  the  date  on  which  this 
book  was  charged  out. 


. 


[30m-6,'ll] 


YB   17453 


NEW  GEOGRAPHIES, 


ZFTJIBLISIEIiEID 


ILSON,  Ifam 


The  Eclectic  Series  of  Geographies,  consisting  of 

I  three  books,  by  A.  VON  STEINWEHR  and  D.  G.  BRIN  i 

No.  1,  The  Primary  Geography,  is  adapted  to  the  use 

of  pupils  commencing  the  study.     The  language  is  simple 

and  clear;  the  definitions  and  descriptions  are  exact;  the 

I  plan  of  the  book  is  natural,  and 4he'  copper-plate  maps  are 

I  wonderfully  clear  and  definite. 

• 

No.  2,  The  Intermediate  Geography,  is  intended  for 

the  use  of  the  higher  classes  in  Graded  Schools,  and  contains 
the  leading  principles  of  the  science,  so  arranged  as  to  give 
correct  ideas  to  pupils,  and,  at  the  same  time,  require  less 
aid  from  the  teacher  than  any  other  book  extant.  This  book 
i  contains  full  instructions  in  map-drawing. 

NO.  3,  The  SchOOl  Geography,  embraces  a  full  Math 
:  ematical,  Political,    and  Physical  description  of  the  Earth, 
and  is  intended    for  the  highest  classes  in  this  branch  of 
|  study.      The  maps,  which  are  the  basis  of  all  geographical 
study,  are  models  of  cleai/,ess.     The  physical    features  of 
each  country  are  full}'  brought  out,  and  it  is  thus  made  pos- 
sible to  teach  Geography  objectively. 


